Technical Improvements in Quantitative Susceptibility Mapping · PDF fileTECHNICAL...

TECHNICAL IMPROVEMENTS IN

QUANTITATIVE SUSCEPTIBILITY MAPPING

TECHNICAL IMPROVEMENTS IN

QUANTITATIVE SUSCEPTIBILITY MAPPING

BY

SAIFENG LIU B.Sc.

A Thesis

Submitted to the School of Biomedical Engineering

and the School of Graduate Studies

of McMaster University

in Partial Fulfilment of the Requirements

for the Degree of

Doctor of Philosophy

© Copyright by Saifeng Liu, April 2014

All Rights Reserved

ii

Ph. D. (2014) McMaster University

(School of Biomedical Engineering) Hamilton, Ontario, Canada

TITLE: Technical Improvements in Quantitative Susceptibility Mapping

AUTHOR: Saifeng Liu

B.Sc., (Biomedical Engineering)

Huazhong University of Science and Technology

Wuhan, China

SUPERVISOR: E. Mark Haacke Ph. D.

NUMBER OF PAGES: xx, 192

iii

To my parents

Fenling Liu and Yongqi Liu

iv

Abstract

Quantitative susceptibility mapping (QSM) is a promising technique to study tissue

properties and function in vivo. The presence of a susceptibility source will lead to a non-

local field variation which manifests as a non-local behavior in magnetic resonance phase

images. QSM is an ill-posed inverse problem that maps the phase back to the

susceptibility source. In practice, the phase images are usually contaminated by

background field inhomogeneities. Consequently, the efficacy and accuracy of QSM rely

on background field removal. In this thesis, several technical advances in QSM have been

made which accelerate the data processing and improve the accuracy of this ill-posed

problem.

Different background field removal algorithms are analyzed and compared in detail,

including homodyne high-pass filtering, variable high-pass filtering, sophisticated

harmonic artefact reduction for phase data (SHARP), and projection onto the dipole field

(PDF). In these algorithms, phase unwrapping is usually required, which can be time-

consuming and sensitive to noise. To solve this problem, a new background field removal

algorithm, local spherical mean value filtering (LSMV), is proposed, in which the global

phase unwrapping is bypassed. This algorithm improves the time-efficiency and

robustness of background field removal, especially for double-echo data.

The simplest algorithm to solve the inverse problem is the regularization process using

truncated k-space division. However, this algorithm induces streaking artefacts in the

v

susceptibility maps. The streaking artefacts can be reduced dramatically using geometry

constraints. In the k-space/image domain iterative algorithm for susceptibility weighted

imaging and mapping (SWIM), the geometries extracted from the initial susceptibility

maps are used to update the data in the singularity regions in k-space. An improved

version of this algorithm is demonstrated using multi-level thresholding to account for the

variation in the susceptibilities of different structures in the brain.

These susceptibility maps could be used to generate orientation independent weighting

masks, to form a new type of susceptibility weighted image (SWI), referred to here as

true-SWI (tSWI). The tSWI data show improved contrast-to-noise ratio (CNR) of the

veins and reduced blooming artefacts due to the strong dipolar phase of microbleeds.

Finally, the accuracy in estimating the susceptibility of a small object is usually hampered

by partial volume effects. In this thesis, it is shown that the effective magnetic moment,

being the product of the apparent volume and the measured susceptibility of the small

object, is constant and can be used to improve the susceptibility quantification, if a priori

information of the volume is available.

In conclusion, the technical improvements presented in this thesis contribute to a better

data processing scheme for QSM, with accelerated data processing by using the LSMV

algorithm for background field removal, reduced streaking artefacts in the susceptibility

maps by using the iterative SWIM algorithm for solving the inverse problem, and

improved accuracy by proper handling of the partial volume effects using volume

constraints.

vi

Acknowledgements

I would like to express my deepest appreciation to my supervisor, Dr. E. Mark Haacke,

for his enduring support and persistent inspiration. It is Dr. Haacke who led me into the

exciting and intriguing world of MRI. As one of the great scientists in this field,

Dr. Haacke has demonstrated to me how the knowledge of spins in MRI and expertise in

physics can be applied to both MRI and tennis. It is his passion for both research and life

that has encouraged me to finish the work presented in this thesis.

I would like to thank my supervisory committee members, Dr. Nicholas Bock,

Dr. Qiyin Fang, and Dr. Maureen MacDonald, for their valuable guidance and comments.

Thanks to Dr. Michael Noseworthy, for helping me on extending my research horizon

and improving my teaching skills.

Moreover, I would like to thank Dr. Yu-Chung Norman Cheng, Dr. Yongquan Ye and

Dr. Jaladhar Neelavalli, for their careful reviewing of my thesis. Thanks to my colleagues

and friends Sagar Buch M. Sc., Jin Tang Ph.D., Eyesha Hashim, B. Sc., Weili Zheng

Ph.D. and Manju Liu M. Sc.

Finally, I wish to thank my grandfather Bingyan Liu (1933-2012) and grandmother

Hongzhu Liu, for their endless love.

vii

Contents

Abstract .............................................................................................................................. iv

Acknowledgements ........................................................................................................... vi

List of Figures .................................................................................................................... ix

List of Tables ................................................................................................................. xvii

List of Abbreviations ................................................................................................... xviii

1 Introduction ..................................................................................................................... 1

1.1 Background and significance ..................................................................................... 1

1.2 Review of QSM techniques ....................................................................................... 4

1.3 Overview of the thesis ............................................................................................... 7

2 Basic Concepts of Phase, Gradient Echo Imaging and Quantitative Susceptibility

Mapping ............................................................................................................................ 13

2.1 The concept of a gradient echo and phase ............................................................... 13

2.2 Predicting field variation through forward calculation ............................................ 20

2.3 Quantifying Susceptibility as an inverse problem ................................................... 23

2.4 Susceptibility and its relations to venous oxygen saturation and iron content ........ 25

3 Background Field Removal .......................................................................................... 29

3.1 The background field ............................................................................................... 29

3.2 Homodyne high-pass filter ....................................................................................... 31

3.3 Variable high-pass filter (VHP) ............................................................................... 37

3.4 Sophisticated Harmonic Artefact Reduction for Phase data (SHARP) ................... 44

3.5 Comparison of different background field removal algorithms............................... 49

4 Fast and Robust Background Field Removal using Double-echo Data ................... 62

4.1 Introduction .............................................................................................................. 62

viii

4.2 Theory ...................................................................................................................... 64

4.3 Materials and Methods ............................................................................................. 67

4.4 Results ...................................................................................................................... 74

4.5 Discussion ................................................................................................................ 85

5 Solving the Inverse Problem of Susceptibility Quantification .................................. 93

5.1 Susceptibility mapping using truncated k-space division ........................................ 93

5.2 Geometry constrained iterative reconstruction ...................................................... 100

5.3 Noise in susceptibility mapping ............................................................................. 120

5.4 Discussion and Conclusions .................................................................................. 124

6 Improved Venography using True Susceptibility Weighted Imaging (tSWI) ...... 129

6.1 Introduction ............................................................................................................ 129

6.2 Materials and Methods ........................................................................................... 131

6.3 Results .................................................................................................................... 139

6.4 Discussion .............................................................................................................. 150

7 Quantitative Susceptibility Mapping of Small Objects using Volume Constraints

.......................................................................................................................................... 157

7.1 Introduction ............................................................................................................ 157

7.2 Theory and Methods .............................................................................................. 159

7.3 Results .................................................................................................................... 169

7.4 Discussion and Conclusions .................................................................................. 177

8 Conclusions and Future Directions ........................................................................... 185

ix

List of Figures

Figure 2.1 Sequence diagram of a single echo 3D gradient echo sequence. GS: slab

selection/partition encoding gradient; GP: phase encoding gradient; GR: readout gradient.

............................................................................................................................................ 16

Figure 2.2 Illustration of filling one line in k-space. The k-space trajectory as indicated

by the dashed arrows is described using Eqs.2.8 to 2.10. .................................................. 17

Figure 2.3 Sequence diagram of a single echo 3D gradient echo sequence with flow-

compensation in all directions. GS: slab selection/partition encoding gradient; GP: phase

encoding gradient; GR: readout gradient. ........................................................................... 19

Figure 2.4 QSM data processing procedures. The dashed line indicates that brain masks

may not be required for phase unwrapping. The * indicates that the phase unwrapping

step can be avoided in certain algorithms and the unwrapped phase is not required for

background field removal. ................................................................................................. 23

Figure 3.1 Illustration of the processing steps in homodyne high-pass filtering. ............. 33

Figure 3.2 Relative error in estimated susceptibilities induced by high-pass filtering in

different orientations. The relative errors induced by homodyne high-pass filtering with

different sizes, for cylinders in the 90o case, are shown in a. b to e show the effects of the

orientation of high-pass filtering for cylinders in the 90o case (b), in the 30

o case (c), in

the 45o case (d), and in the 60

o case (e). Relative errors for the spheres are shown in f. .. 36

Figure 3.3 a). Simulated phase image with background phase but without random noise.

b). Simulated phase with background phase and random noise. c). The local phase

information without random noise. This is used as the true answer to evaluate the

accuracies of the processed phase images. d). The reference region used for calculating

the RMSEs in the processed phase images. e). The susceptibility map in axial view. f).

The phase image corresponds to e. .................................................................................... 40

Figure 3.4 Overall RMSEs of the processed phase images (a) and susceptibility maps (b)

generated using VHP. ........................................................................................................ 42

Figure 3.5 Relative errors in the estimated susceptibilities of different structures using

phase images processed by VHP. a) Veins. b) Globus pallidus. c) Putamen. d) Caudate.

x

The dashed lines indicate the errors in estimated susceptibilities for different structures

using the phase images with ideal background field removal. .......................................... 43

Figure 3.6 a) Overall RMSEs for different radii of the spherical kernel and different

values of th in SHARP. b) Overall RMSE in susceptibility quantification for different

kernel sizes and different thresholds in SHARP. ............................................................... 46

Figure 3.7 Relative errors in measured susceptibilities using different parameters in

SHARP for different structures. ......................................................................................... 47

Figure 3.8 The minimal RMSEs for different structures at different radii of the spherical

kernel in SHARP. ............................................................................................................... 48

Figure 3.9 Susceptibility estimated using the original phase vs. the susceptibility

estimated using different phase processing methods: a) Homodyne HP32×32, b)

Homodyne HP64×64, c) SHARP (radius=8px, th=0.02), d) SHARP (radius=8px,

th=0.05), e) Variable HP, and f) PDF. ............................................................................... 54

Figure 3.10 Phase images processed with different algorithms in Dataset 1. a and e.

64x64 Homodyne high-pass filter. b and f: VHP. c and g: SHARP. d and h: PDF. ......... 56

Figure 3.11 Susceptibility maps generated with different algorithms in Dataset 1. a and e.

64×64 Homodyne high-pass filter. b and f: VHP. c and g: SHARP. d and h: PDF. ......... 56

Figure 3.12 The means and standard deviations of the measured susceptibility values for

different structures in different in vivo datasets. The error bars represent the standard

deviations of the measured susceptibilities in different datasets. GP: globus pallidus, PUT:

putamen, CN: caudate nucleus, SN: substantia nigra, RN: red nucleus, and THA:

thalamus. The asterisks indicate Tukey’s HSD significance: * p<0.05, ** p<0.01, ***

p<0.001. ............................................................................................................................. 57

Figure 4.1 Processing steps in LSMV for single echo phase data with short TE. ............ 72

Figure 4.2 RMSEs of the processed phase images (a and b) and the susceptibility maps (c

and d) at different noise levels for the simulated 3D brain model. The RMSEs in a and c

were calculated using all the pixels inside the brain, while the RMSEs in b and d were

calculated using only the pixels close to (or inside) the veins. The SNR represents the

signal-to-noise ratio in the magnitude images. .................................................................. 75

Figure 4.3 A comparison of the processing times using different algorithms at different

noise levels. ........................................................................................................................ 76

Figure 4.4 RMSEs in the processed phase images (a and b) and susceptibility maps (c

and d) of the cylinders at two TEs. The SNR represents the signal-to-noise ratio in the

magnitude images. ............................................................................................................. 77

xi

Figure 4.5 The original phase images and the local phase images generated using

different algorithms for the simulated cylinder at SNR=5:1 and SNR=20:1. The SNR

represents the signal-to-noise ratio in the magnitude images. The images in the second to

fourth columns are the central 64×64 pixels in the processed local phase images, as

indicated by the white dashed box in the top-left image. The errors in the local phase

images obtained using 3DSRNCP and Laplacian phase unwrapping are indicated by the

black arrows. ...................................................................................................................... 78

Figure 4.6 The original and processed phase images for Dataset 1. a) Original phase

image at TE1=7.38ms. b) Complex divided phase image with effective TE=2.84ms. c)

Original phase images at TE2=17.6ms. d) SMV filtered result (φSMV) at TE1. e) SMV

filtered result at ΔTE. f) SMV filtered result at TE2, calculated using the images shown in

e and d as e+2×d. g) Local phase image at TE1. h) Local phase image at ΔTE. i) local

phase image at TE2. ............................................................................................................ 79

Figure 4.7 Comparison between the local phase images and susceptibility maps generated

using LSMV (a, d and g) and those generated using 3DSRNCP (b, e and h) for Dataset 1.

The difference images are shown in c, f and i. The phase images in the first row are at

TE1, and the phase images in the second row are at TE2. The images in the third row are

susceptibility maps (SM) obtained using the phase images at TE2 (d and e). The scale bars

are for the difference images c, f and i only. Note the improvement in the SM using

LSMV thanks to the better recovery of phase around the veins. ....................................... 80


using LSMV (a, d and g), and those generated using Laplacian phase unwrapping (b, e

and h) for Dataset 1. The difference images are shown in c, f and i. The phase images in

the first row are at TE1, and the phase images in the second row are at TE2. The images in

the third row are susceptibility maps obtained using the phase images at TE2 (d and e).

The scale bars are for the difference images c, f and i only. .............................................. 81

Figure 4.9 a) Original phase image at TE2 from Dataset 2 with cusp artefact. Note that,

this image was obtained using the built-in multi-channel data combination algorithm on

the scanner. b) Unwrapped phase image using 3DSRNCP. c) Unwrapped phase image

using Laplacian phase unwrapping. d) Local phase image generated using LSMV. e)

Local phase image generated using the unwrapped phase image shown in b. f) Local

phase image obtained using the unwrapped phase image shown in c. Cusp artefact caused

errors in the processed phase images, as indicated by the white arrows. .......................... 84

Figure 5.1 The cross-sections of (a and b) and (c and d). In a and c, the

cross-sections are parallel to the main field direction, in b and d, the cross-sections are

perpendicular to the main field direction. The white regions in a and b correspond to the

xii

regions where . The black dashed lines in a and c indicate the positions of

the cross-sections shown in b and d................................................................................... 94

Figure 5.2 The underestimation in the susceptibility values measured inside the cylinders

with different radii (a, c and e) and the standard deviations (b, d and f). a and b were

generated when no noise was added. For c and d, SNR=10:1 in the magnitude images;

while for e and f, SNR=5:1. ............................................................................................... 98

Figure 5.3 The mean susceptibility values (a, c and e) and the standard deviations (b, d

and f) measured in the background regions outside the cylinders. a and b were generated

when no noise was added. For c and d, SNR=10:1 in the magnitude images; while for e

and f, SNR=5:1. ................................................................................................................. 99

Figure 5.4 The cost function R(th) for different cylinders. R(th) was calculated using Eq.

5.7. The optimal th was determined as when R(th) was minimized. Particularly, when

SNR=10:1, the optimal ths were 0.12 (r=2px), 0.13(r=4px), 0.14 (r=8px), and 0.14

(r=16px). When SNR=5:1, the optimal ths were 0.1 (r=2px), 0.13(r=4px), 0.13 (r=8px),

and 0.14 (r=16px). ............................................................................................................ 100

Figure 5.5 Illustration of the processing steps of Iterative SWIM algorithm. ................ 103

Figure 5.6 The effects of the parameter th on the accuracies of the estimated

susceptibilities of different structures. a) The relative errors (absolute values) in the

measured mean susceptibilities at different values of th. b) The standard deviations of the

measured susceptibilities at different values of th. .......................................................... 106

Figure 5.7 Means and standard deviations of the susceptibility values of different

structures in the 3D brain model at different iteration steps. a and b: the mean

susceptibility values measured at different iteration steps. c to f: the standard deviations

measured at different iteration steps. The images in the first column (a, c and e) show the

changes in mean and standard deviation when a high χth was used in geometry extraction,

while the images in the second column show the results when a low χth was used in

geometry extraction. ........................................................................................................ 107

Figure 5.8 a) The initial susceptibility map. b) The final susceptibility map after iterative

SWIM. c) The difference between a and the ideal susceptibility map. d) The difference

between b and the ideal susceptibility map. .................................................................... 108

Figure 5.9 K-space profiles of the initial susceptibility map (a) and the final susceptibility

map after iterative SWIM algorithm (b). The errors in the k-space profiles in a and b are

shown in c and d, respectively. Specifically, the errors were calculated by comparing the

k-space of the generated susceptibility maps with the kspace of the ideal susceptibility

maps. Compared with the initial susceptibility maps, the final susceptibility maps after

xiii

iterative SWIM algorithm have reduced errors in the cone of singularities in k-space, as

indicated by the white arrows. ......................................................................................... 109

Figure 5.10 The extracted geometries of veins and grey matter structures (a and b) and

the susceptibility maps (c and d) for the in vivo data. a) Maximum intensity projection of

the binary masks of veins extracted using a high χth. b) One slice of the binary masks of

the grey matter structures extracted using a low χth. c) Maximum intensity projection of

the initial susceptibility maps. d) The corresponding susceptibility map to the binary

mask shown in b. ............................................................................................................. 110

Figure 5.11 Mean susceptibility values of veins and globus pallidus at different iteration

steps in the in vivo data. The figures in the first column were obtained when high χth was

used to extract the geometries of veins only, while the figures in the second column were

measured when low χth was used to extract the geometries of veins and other structures.

.......................................................................................................................................... 112

Figure 5.12 Mean susceptibility values of red nucleus (RN), caudate nucleus (CN) and

putamen (PUT) at different iteration steps in the in vivo data. The figures in the first

column were obtained when high χth was used to extract the geometries of veins only,

while the figures in the second column were measured when low χth was used to extract

the geometries of veins and other structures. ................................................................... 113

Figure 5.13 Standard deviations of the susceptibility values at different iteration steps for

different structures in the in vivo data. The figures in the first column were obtained when

high χth was used to extract the geometries of veins only, while the figures in the second

column were measured when low χth was used to extract the geometries of veins and other

structures. ......................................................................................................................... 114

Figure 5.14 Comparison between the initial susceptibiltiy maps (a and d) and the final

susceptibility maps (b and e). Their differences are shown in c and f (generated as b-a

and e-d). ........................................................................................................................... 115

Figure 5.15 The effects of the regularization parameter λ in MEDI. The average gradient

was measured in the low gradient regions in the converged susceptibility maps. ........... 116

Figure 5.16 Susceptibility maps generated using different values of regularization

parameter λ with MEDI. a and d: λ=10. b and e: λ=100. a and d: λ=1000. ................... 116

Figure 5.17 Comparison of susceptibility maps generated by iterative SWIM (a) and

MEDI (b). The profiles of the solid black lines across the vein of Galen in a and b are

shown in c. The profiles of the dashed black line in the cortical region were shown in d.

The corresponding magnitude image is shown in e, which does not have clear edge

information of the veins in the region indicated by the white arrow. .............................. 117

xiv

Figure 5.18 Distributions of the susceptibility values of the pixels inside the veins (a and

b) and inside the globus pallidus (c and d), measured from iterative SWIM generated

susceptibility maps (a and c) and from MEDI generated susceptibility maps (b and d). 118

Figure 5.19 The k-space profiles obtained using truncated k-space division (a), iterative

SWIM (b) and MEDI (c). The difference between b and c is shown in d. The window

levels in a, b, and c are the same. The major differences are seen along the cone of

singularities in k-space, as indicated by the white arrow. ................................................ 120

Figure 5.20 a) Standard deviation of susceptibilities measured in the background

reference region for cylinders with different radii. b) Standard deviation measured inside

the cylinders. c) Noise amplification factor α as a function of SNR in magnitude images,

when th=0.1. d) Noise amplification factor α as a function of relative errors in the

estimated susceptibilities, when SNR=10:1. The arrow shows the case when th=0.13,

α=3.3 and the underestimation of the susceptibility is 11.3%. ........................................ 123

Figure 6.1 A comparison between tSWI and SWI data processing steps. ...................... 132

Figure 6.2 Phase images (a and b), susceptibility maps (c and d) and tSWI images (e, f, g

and h) for simulated cylinders with and without homodyne high-pass filtering. Images in

the first and third columns are generated using the original phase images without high-

pass filtering, while images in the second and fourth columns are generated using high-

pass filtering. The tSWI images e and f were generated using χ1=0, χ2=0.45ppm, n=2;

while g and h were generated using χ1=3σχ, χ2=0.45ppm, n=4. σχ is the standard deviation

of a reference region measured from the susceptibility maps shown in c and d

(σχ=0.05ppm for both c and d). The SNR in the original magnitude image was set to be

10:1 and the CNR between the cylinders and background in the original magnitude

images was basically zero. ............................................................................................... 140

Figure 6.3 Measured CNRs of cylinders from simulated tSWI images. Figures in

different rows were generated using different χ2 values, while figures in different columns

were generated using different χ1 values. a) χ1=0, χ2=1ppm; b) χ1=3σχ, χ2=1ppm; c) χ1=0,

χ2=0.45ppm; d) χ1=3σχ, χ2=0.45ppm; e) χ1=0, χ2=0.45ppm; and f) χ1=3σχ, χ2=0.45ppm.

To evaluate the effect of high-pass filtering, e and f were generated using high-pass

filtered phase images. ....................................................................................................... 141

Figure 6.4 Theoretically predicted CNRs of cylinders with different susceptibility values.

Figures in different rows were generated using different χ2 values, while figures in

different columns were generated using different χ1 values. a) χ1=0, χ2=1ppm; b) χ1=3σχ,

χ2=1ppm; c) χ1=0, χ2=0.45ppm; and d) χ1=3σχ, χ2=0.45ppm. ......................................... 142

Figure 6.5 Local CNRs of the right septal vein (a, c, and e) and the left internal cerebral

vein (b, d, and f) from different datasets with isotropic resolution. a, b, c and d were

xv

generated when threshold χ1=0 was used to create the susceptibility weighting masks,

while e and f were generated when χ1=3σχ was used. The CNRs were normalized by the

corresponding SNRs listed in Table 6.2. c and d show the CNRs of the two veins in

Dataset 2 with isotropic resolution, when different data processing methods were used for

susceptibility mapping (see Fig. 6.7 for examples of the tSWI images). ........................ 144


vein (b, d, and f) from different datasets with anisotropic resolution. a, b, c and d were




Dataset 2 with anisotropic resolution, when different data processing methods were used

for susceptibility mapping (see Fig. 6.7 for examples of the tSWI images).................... 145

Figure 6.7 Comparison between minimal intensity projections (mIP) of tSWI and SWI

data over 16mm for isotropic (top row) and anisotropic data (bottom row) for Dataset 2.

For b, c, f and g, susceptibility maps were generated using homodyne high-pass filtering

and thresholded k-space division; while for d and h, susceptibility maps were generated

using SHARP and geometry constrained iterative algorithm. a) isotropic SWI mIP; b)

isotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2); c) isotropic tSWI mIP (χ1=3σχ, χ2=0.45ppm,

n=4); d) isotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2); e) anisotropic SWI mIP. f)

anisotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2). g) anisotropic tSWI mIP (χ1=3σχ,

χ2=0.45ppm, n=8). h) anisotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2). ......................... 147

Figure 6.8 A sagittal view showing a vein near the magic angle (54.7° relative to the

main magnetic field) as indicated by the black arrows. a) Phase image (from a left-handed

system) showing effectively zero phase inside the vein, with outer field dipole effects also

visible; b) susceptibility maps showing the vein as uniformly bright; c) susceptibility

weighting mask obtained from the phase image (n=4); d) susceptibility weighting mask

obtained from the susceptibility maps (χ1=0, χ2=0.45ppm, n=2); e) SWI showing

unsuppressed signal inside the vein; and f) tSWI showing a clear suppression of the vein

even at the magic angle. g) mIP of SWI in the sagittal direction. h) mIP of tSWI in the

sagittal direction. Note the vessels near the magic angle are now well delineated in the

tSWI data. ........................................................................................................................ 149

Figure 6.9 Sagittal views of SWI (a and c) and tSWI images (b and d) in a TBI case. The

microbleeds appear much bigger on the SWI images than on the tSWI images, as

indicated by the white arrows. This is due to the non-local phase information used in the

conventional SWI weighting mask. For better visualization, the images were interpolated

in through-plane direction from a resolution of 0.5mm×0.5mm×2mm to 0.5mm isotropic

resolution. ......................................................................................................................... 150

xvi

Figure 7.1 Apparent volume normalized to the volume at TE = 20 ms (first column),

measured susceptibility (second column), and normalized magnetic moments (third

column) measured at different TEs of four different spheres. The dashed lines in the

second column (b, e, h and k) indicate the true susceptibility 9.4 ppm. For each sphere,

the effective magnetic moments were normalized to the true effective magnetic moment.

.......................................................................................................................................... 171

Figure 7.2 Axial, sagittal and coronal views of the susceptibility maps with TE=3.93ms

(a, b and c) and TE=26.61ms (d, e and f). The main field direction is in “y” direction.

Glass bead No. 9 in Table 7.1 is pointed by the white arrows. The air bubbles are pointed

by the white dashed arrows. ............................................................................................. 174


(a, b and c) and TE=26.61ms (d, e and f), obtained using newer data processing

algorithms. ....................................................................................................................... 174

Figure 7.4 a) Originally measured susceptibility values at different TEs for glass beads

and air bubbles. b) Corrected susceptibility values. c) Distribution of the originally

measured susceptibility values. d) Distribution of the corrected susceptibility values.

After correction using the spin echo volume, the glass beads can be clearly distinguished

from air bubbles. These results were obtained using the new data processing algorithms.

.......................................................................................................................................... 177

xvii

List of Tables

Table 3.1 Imaging parameters for the in vivo data. ........................................................... 51

Table 3.2 Data processing parameters in different algorithms. ......................................... 51

Table 3.3 The estimated susceptibilities (mean ± std. in ppm) for different structures in

the brain model using different phase processing methods. “Original Phase” represents

using the simulated phase images without any background field. ..................................... 52

Table 4.1 Imaging parameters for the in vivo data. Datasets 1 and 2 were collected on the

same volunteer. .................................................................................................................. 70

Table 4.2 A comparison of phase images and susceptibility maps processed using

different algorithms for the in vivo data. ............................................................................ 83

Table 5.1 Mean and standard deviation (in ppm) measured from susceptibility maps

generated using iterative SWIM algorithm and MEDI. ................................................... 119

Table 6.1 Imaging parameters for three volunteers and one patient for in vivo studies.

Dataset 5 was collected on a TBI patient. ........................................................................ 137

Table 7.1 Spin echo volume (in voxels) and the diameter (in mm) calculated from spin

echo volume for each glass bead. .................................................................................... 173

Table 7.2 Spin echo volume (in voxel) of the 14 air bubbles. ........................................ 173

Table 7.3 Mean measured and corrected susceptibilities (in ppm) of the glass beads and

air bubbles and different TEs. .......................................................................................... 176


air bubbles and different TEs. These results were obtained using the new data processing

algorithms. ....................................................................................................................... 176

xviii

List of Abbreviations

ANOVA Analysis of variance

BW Bandwidth

CMRO2 Cerebral metabolic rate of oxygen

CN Caudate nucleus

CNR Contrast-to-noise ratio

CSF Cerebrospinal fluid

FA Flip angle

FOV Field of view

FSL FMRIB Software Library

GDAC Geometry dependent artefact correction

GP Globus pallidus

GRAPPA Generalized Autocalibrating Partially Parallel Acquisitions

GRE Gradient recalled echo

Hct Hematocrit

ICP-MS Inductively coupled plasma mass spectrometry

Iterative SWIM k-space/image domain iterative algorithm for susceptibility

weighted imaging and mapping

LSMV Local spherical mean value filtering

xix

MEDI Morphology enabled dipole inversion

mIP Minimum intensity projection

MIP Maximum intensity projection

MRI Magnetic resonance imaging

MS Multiple sclerosis

PD Parkinson’s disease

PDF Projection onto the Dipole Field

PRELUDE Phase region expanding labeller for unwrapping discrete

estimates

PUT Putamen

QSM Quantitative susceptibility mapping

RMSD Root-mean-square-deviation

RMSE Root-mean-square-error

RN Red nucleus

ROI Region of interest

SHARP Sophisticated Harmonic Artifact Reduction for Phase data

SMV Spherical mean value filtering

SN Substantia Nigra

SNR Signal-to-noise ratio

SSS Superior sagittal sinus

SWI Susceptibility weighted imaging

TBI Traumatic brain injury

TE Echo Time

THA Thalamus

xx

TR Repetition time

tSWI True susceptibility weighted imaging

VHP Variable high-pass filter

VOI Volume of interest

XRF X-ray fluorescence

Ph.D. Thesis – Saifeng Liu McMaster – School of Biomedical Engineering

1

Chapter 1 Introduction

1.1 Background and significance

Magnetic Resonance Imaging (MRI) provides both structural and functional information

through magnitude and phase images. The contrast in the images is dependent on the

sequence design and the associated acquisition parameters. In Susceptibility Weighted

Imaging (SWI), phase images are combined with magnitude images to enhance the

visualization of veins, iron (in the form of ferritin or hemosiderin if microbleeds have

occurred) or calcium (1-3). Although SWI has been widely used for many clinical

applications, it is hampered by the orientation dependence of phase information,

especially when high imaging resolution is used (4,5). In that case, there will be errors in

visualizing the veins and microbleeds. Besides, the orientation dependence of phase

makes it difficult to quantify the iron/calcium content using SWI. On the other hand,

susceptibility is known to be (for the most part) independent of orientation (5,6). Thus,

mapping the susceptibility distribution, the source of phase information, is of great

interest.

Quantitative Susceptibility Mapping (QSM) provides a robust means to elucidate tissue

properties and function through tissue susceptibilities, which are related to the changes in


2

the deposition of paramagnetic (e.g., iron) or diamagnetic (e.g., calcium or myelin content

in white matter) substances. Susceptibility changes are also related to the changes in the

oxygenation level of venous blood. As a result, QSM has many potential clinical

applications, such as the quantification of cerebral iron deposition or calcium (7),

visualization and quantification of iron loaded biomarkers such as iron loaded stem cells,

as well as quantification of venous oxygen saturation (8,9). In this section, we shall give a

brief review of these applications.

Cerebral iron content can be categorized as heme iron and non-heme iron (16). The

former is related to hemoglobin and transportation of oxygen, while the latter is related to

iron storage or deposition, predominantly in the form of ferritin and hemosiderin

macromolecules which are paramagnetic. Excessive iron deposition in deep gray matter

structures such as the basal ganglia has been observed in many neurodegenerative

diseases including: Alzheimer’s disease (10), Parkinson’s disease (11,12), and multiple

sclerosis (13-16) to name just a few. Increased iron deposition is also present in the

normal aging process (17-19). Studies have been performed to investigate the relation

between the measured susceptibility and the absolute iron content using in vitro ferritin

solutions (20). This relation is further compared and validated with that found in cadaver

brains, for which the iron content can be measured by both QSM and other quantitative

methods such as inductively coupled plasma mass spectrometry (ICP-MS) and x-ray

fluorescence (XRF) (20,21). While the quantification of cerebral iron deposition,

especially in the basal ganglia structures, helps to monitor the progress of

neurodegenerative diseases and to evaluate treatment, a temporal profile and normal


3

baseline of the cerebral iron deposition in the normal aging process would facilitate the

discrimination of the subjects with excessive cerebral iron deposition from normal

healthy controls. This could be particularly beneficial for the early diagnosis of

neurodegenerative diseases.

Using QSM, it is viable to quantify not only the paramagnetic substances, but also

diamagnetic substances such as calcium. Detection and quantification of calcium has long

been a topic of interest in MRI. For example, SWI was formerly used to detect

calcifications in the breast (3). QSM has also been used to differentiate between

diamagnetic and paramagnetic cerebral lesions (7). Moreover, measuring calcification in

vessel walls is also of great interest. For example, intracranial arterial calcification has

been shown to be highly correlated with coronary artery disease for ischemic stroke

patients (22).

Another potential application of QSM is to monitor the cerebral

myelination/demyelination process. This may be useful in studying demyelinating

diseases such as multiple sclerosis and acute disseminated encephalomyelitis (23). It has

been shown that the diamagnetic myelin is the major source of susceptibility differences

between white matter and gray matter including phase and T2*/R2

* effects (24,25). This

has been shown by comparing the grey and white matter phase contrast between

demyelinated shiverer mice and normal control mice using MRI, followed by histological

staining of myelin. The contribution of myelin content to grey/white matter phase contrast

was also validated using post-mortem studies (26). Meanwhile, orientation dependence of

T2*/R2

* was observed and was attributed to the fibre orientation of myelin content in


4

white matter (25,27,28). This orientation dependence of T2* was the strongest in the optic

radiation, which is known to have little iron content (27). Susceptibility anisotropy was

also observed in the deep white matter and a susceptibility tensor model has been invoked

in an attempt to describe the susceptibility behaviour found in the white matter (6).

Finally, the measurement of venous oxygen saturation is another important application of

QSM. It has been shown that the changes in venous oxygen saturation can be reflected by

the changes in susceptibility(8). Together with the flow information, MRI can be used to

quantify cerebral metabolic rate of oxygen (CMRO2) and to examine the cerebral

functional changes in stroke and other neurodegenerative diseases. For example, it was

shown that the visibility of periventricular veins was reduced in multiple sclerosis patients,

due to the reduced brain function and reduced utilization of oxygen (hence more

diamagnetic venous blood) (29). Furthermore, it is of great interest to measure the venous

oxygen saturation in the spinal cord, in order to study the mechanism of blood flow and

oxygen regulation (30,31). All of these applications discussed above depend on proper

reconstruction of the susceptibility map.

1.2 Review of QSM techniques

The past few years have witnessed much progress in QSM. Various in vivo data

acquisition and processing methods have been proposed. The in vivo MR data are

typically acquired using a gradient echo (GRE) sequence with either single or multiple

echoes (8,32,33). Accurate quantification of the susceptibility relies on many factors. The

choice of echo time (TE) will affect the phase image signal-to-noise ratio (SNR) and the


5

level of phase aliasing (8). Meanwhile, T2* signal decay and the blooming artefact at long

TEs may lead to errors in susceptibility quantification (34,35) due to aliasing of the phase

at the edge of the structure of interest. Another important parameter in data acquisition is

the resolution. While low resolution leads to more severe partial volume effects and hence

a larger error in susceptibility quantification, especially for small objects (35), high

resolution generally requires a longer scan time and has reduced SNR. Given the

development in fast imaging methods (36,37), it can be expected that data acquired with

high resolution will become more and more common in the future. Furthermore, when the

focus is the veins, full multi-directional flow compensation is required in order to avoid

the spurious phase component induced by blood flow (mostly in the arteries) (5,8,38).

In QSM data processing, the two most important steps are background field removal and

the inverse process used to reconstruct the susceptibility map from the local phase

information. QSM relies on pristine field (phase) information, which is induced by the

local susceptibility distribution. The background field, on the other hand, is mainly

induced by the global geometry such as the air/bone-tissue interfaces and main magnetic

field inhomogeneity (2,38-40). Background fields lead to phase aliasing and signal decay

at a long TE. Typically, the background field is removed through high-pass filtering, due

to its low spatial frequency (2). But high-pass filtering causes inevitable signal loss of the

local field variation. For a particular object, the signal loss due to high-pass filtering is

dependent on both the size of the high-pass filter and the size of the object. Newer

algorithms aim to reduce this signal loss of the local field while effectively removing the

background field (38,40). One problem with all these algorithms is the loss of information


6

near the interface between susceptibility sources, such as the edge of the brain and air.

Normally an eroded binary mask is used to perform the calculation only for the central

regions (38,40). Another common problem is the requirement of phase unwrapping

(38,40). Phase unwrapping is sensitive to noise in phase images and is particularly

problematic when cusp artefacts are present (41,42). Cusp artefacts are typically caused

by an improper combination of multi-channel phase data (1,43,44). Further, multi-

dimensional phase unwrapping is usually time-consuming (45,46). Fast phase

unwrapping algorithms such as the Laplacian based algorithm can still suffer from errors

in regions with rapid field changes, i.e., the edges of the veins or air/tissue interfaces

(41,47). The use of double or multi-echo data helps to alleviate the problems related to

phase unwrapping (48). Particularly, when a double-echo GRE sequence with a short first

TE is used, phase unwrapping can be avoided in the background field removal step, an

approach that will be taken later in this thesis.

Using the extracted local phase information, various algorithms have been proposed to

reconstruct the susceptibility map, based on the relation between the susceptibility

distribution and local field variation. However, the Green’s function in the Fourier

domain has zeros and thus QSM is an ill-posed inverse problem (8,32,49-52). The

singularities of the Green’s function lead to streaking artefacts in susceptibility maps even

when regularization methods are used. The simplest way of solving this inverse problem

is to define a Fourier domain threshold and to use truncated k-space division (8,53). A

larger threshold leads to a reduced level of streaking artefacts but also a larger error in the

estimated susceptibility due to more signal loss in k-space (8) and more blurring of


7

individual structures (41). There is always this type of trade-off between the accuracy in

the susceptibility estimation and the quality of artefact suppression.

In conclusion, the robustness of QSM processing is usually hampered by the background

field removal step. Particularly, the requirement of phase unwrapping in background field

removal makes it vulnerable to cusp artefacts in phase images. The accuracy of QSM

relies on several factors. Among those is the geometry information which plays an

indispensable role. The motivation of this study is to develop fast and robust data

processing methods for quantitative susceptibility mapping and to study the accuracy in

susceptibility estimation.

1.3 Overview of the thesis

In Chapter 2, we introduce the basic theories, including signal formation mechanisms

using a gradient echo sequence, the phase information and magnetic susceptibility. Given

an arbitrary susceptibility distribution, the induced field variation can be predicted

through a forward field calculation. On the other hand, QSM requires solving an inverse

problem. An overview of QSM data processing procedures is given in that chapter. In

Chapter 3, we focus on the first important step in QSM data processing: background field

removal. Different filtering techniques are discussed in detail, including homodyne high-

pass filtering, variable high-pass filtering (VHP), Sophisticated Harmonic Artefact

Reduction for Phase data (SHARP) and Projection onto Dipole Fields (PDF). The

accuracy of these filters are evaluated and compared. In Chapter 4, a double-echo phase

processing algorithm is introduced. The advantages of such an algorithm are its


8

robustness and time-efficiency. Chapter 5 is focused on the core procedure of QSM, the

inverse process. A geometry constrained iterative algorithm for susceptibility mapping is

introduced and evaluated. A direct application of QSM is to use the susceptibility map to

generate susceptibility weighting masks to improve the visualization of the venous

structures, as is discussed in Chapter 6. Chapter 7 deals with the limitation of

susceptibility estimation of small objects. In that chapter, we show that even though the

accuracy of the susceptibility estimation is limited by the apparent volume, the product of

susceptibility and volume (or the effective magnetic moment) is constant. Conclusions

and future directions are provided in Chapter 8.


9

References

1. Haacke EM, Mittal S, Wu Z, Neelavalli J, Cheng YCN. Susceptibility-Weighted

Imaging: Technical Aspects and Clinical Applications, Part 1. Am. J. Neuroradiol.

2009;30:19-30.

2. Haacke EM, Xu Y, Cheng YCN, Reichenbach JR. Susceptibility weighted imaging

(SWI). Magn. Reson. Med. 2004;52:612-8.

3. Fatemi-Ardekani A, Boylan C, Noseworthy MD. Identification of breast

calcification using magnetic resonance imaging. Med. Phys. 2009;36:5429-36.

4. Xu Y, Haacke EM. The role of voxel aspect ratio in determining apparent vascular

phase behavior in susceptibility weighted imaging. Magn. Reson. Imaging

2006;24:155-60.

5. Haacke EM, Reichenbach JR, editors. Susceptibility Weighted Imaging in MRI:

Basic Concepts and Clinical Applications. 1st ed. Wiley-Blackwell; 2011.

6. Liu C. Susceptibility tensor imaging. Magn. Reson. Med. 2010;63:1471-7.

7. Schweser F, Deistung A, Lehr BW, Reichenbach JR. Differentiation between

diamagnetic and paramagnetic cerebral lesions based on magnetic susceptibility

mapping. Med. Phys. 2010;37:5165.

8. Haacke EM, Tang J, Neelavalli J, Cheng YC. Susceptibility mapping as a means to

visualize veins and quantify oxygen saturation. J. Magn. Reson. Imaging

2010;32:663-76.

9. Jain V, Abdulmalik O, Propert KJ, Wehrli FW. Investigating the magnetic

susceptibility properties of fresh human blood for noninvasive oxygen saturation

quantification. Magn. Reson. Med. 2012;68:863-7.

10. Langkammer C, Ropele S, Pirpamer L, Fazekas F, Schmidt R. MRI for Iron

Mapping in Alzheimer’s Disease. Neurodegener. Dis. 2013;

11. Wallis LI, Paley MNJ, Graham JM, Grünewald RA, Wignall EL, Joy HM, et al.

MRI assessment of basal ganglia iron deposition in Parkinson’s disease. J. Magn.

Reson. Imaging 2008;28:1061-7.

12. Wang Y, Butros SR, Shuai X, Dai Y, Chen C, Liu M, et al. Different iron-deposition

patterns of multiple system atrophy with predominant parkinsonism and idiopathetic

Parkinson diseases demonstrated by phase-corrected susceptibility-weighted

imaging. AJNR Am. J. Neuroradiol. 2012;33:266-73.

13. Al-Radaideh AM, Wharton SJ, Lim S-Y, Tench CR, Morgan PS, Bowtell RW, et al.

Increased iron accumulation occurs in the earliest stages of demyelinating disease:

an ultra-high field susceptibility mapping study in Clinically Isolated Syndrome.

Mult. Scler. J. 2013;19:896-903.

14. Haacke EM, Makki M, Ge Y, Maheshwari M, Sehgal V, Hu J, et al. Characterizing

iron deposition in multiple sclerosis lesions using susceptibility weighted imaging. J.

Magn. Reson. Imaging 2009;29:537-44.

15. Langkammer C, Liu T, Khalil M, Enzinger C, Jehna M, Fuchs S, et al. Quantitative

susceptibility mapping in multiple sclerosis. Radiology 2013;267:551-9.

16. Walsh AJ, Lebel RM, Eissa A, Blevins G, Catz I, Lu J-Q, et al. Multiple sclerosis:

validation of MR imaging for quantification and detection of iron. Radiology

2013;267:531-42.


10

17. Haacke EM, Cheng NYC, House MJ, Liu Q, Neelavalli J, Ogg RJ, et al. Imaging

iron stores in the brain using magnetic resonance imaging. Magn. Reson. Imaging

2005;23:1-25.

18. Bilgic B, Pfefferbaum A, Rohlfing T, Sullivan EV, Adalsteinsson E. MRI estimates

of brain iron concentration in normal aging using quantitative susceptibility mapping.

NeuroImage 2012;59:2625-35.

19. Haacke EM, Ayaz M, Khan A, Manova ES, Krishnamurthy B, Gollapalli L, et al.

Establishing a baseline phase behavior in magnetic resonance imaging to determine

normal vs. abnormal iron content in the brain. J. Magn. Reson. Imaging

2007;26:256-64.

20. Zheng W, Nichol H, Liu S, Cheng Y-CN, Haacke EM. Measuring iron in the brain

using quantitative susceptibility mapping and X-ray fluorescence imaging.

NeuroImage 2013;78:68-74.

21. Langkammer C, Krebs N, Goessler W, Scheurer E, Ebner F, Yen K, et al.

Quantitative MR Imaging of Brain Iron: A Postmortem Validation Study. Radiology

2010;257:455-462.

22. Ahn SS, Nam HS, Heo JH, Kim YD, Lee S-K, Han KH, et al. Ischemic stroke:

measurement of intracranial artery calcifications can improve prediction of

asymptomatic coronary artery disease. Radiology 2013;268:842-9.

23. Chitnis T. Pediatric demyelinating diseases. Contin. Minneap. Minn 2013;19:1023-

45.

24. Liu C, Li W, Johnson GA, Wu B. High-field (9.4 T) MRI of brain dysmyelination

by quantitative mapping of magnetic susceptibility. NeuroImage 2011;56:930-8.

25. Lee J, Shmueli K, Kang B-T, Yao B, Fukunaga M, van Gelderen P, et al. The

contribution of myelin to magnetic susceptibility-weighted contrasts in high-field

MRI of the brain. NeuroImage 2012;59:3967-75.

26. Langkammer C, Krebs N, Goessler W, Scheurer E, Yen K, Fazekas F, et al.

Susceptibility induced gray-white matter MRI contrast in the human brain.

NeuroImage 2012;59:1413-9.

27. Sati P, Silva AC, van Gelderen P, Gaitan MI, Wohler JE, Jacobson S, et al. In vivo

quantification of T₂ anisotropy in white matter fibers in marmoset monkeys.

NeuroImage 2012;59:979-85.

28. Lee J, van Gelderen P, Kuo L-W, Merkle H, Silva AC, Duyn JH. T2*-based fiber

orientation mapping. NeuroImage 2011;57:225-34.

29. Ge Y, Zohrabian VM, Osa E-O, Xu J, Jaggi H, Herbert J, et al. Diminished visibility

of cerebral venous vasculature in multiple sclerosis by susceptibility-weighted

imaging at 3.0 Tesla. J. Magn. Reson. Imaging 2009;29:1190-4.

30. Fujima N, Kudo K, Terae S, Ishizaka K, Yazu R, Zaitsu Y, et al. Non-invasive

measurement of oxygen saturation in the spinal vein using SWI: quantitative

evaluation under conditions of physiological and caffeine load. NeuroImage

2011;54:344-9.

31. Fujima N, Kudo K, Terae S, Hida K, Ishizaka K, Zaitsu Y, et al. Spinal

arteriovenous malformation: evaluation of change in venous oxygenation with

susceptibility-weighted MR imaging after treatment. Radiology 2010;254:891-9.


11

32. Liu J, Liu T, de Rochefort L, Ledoux J, Khalidov I, Chen W, et al. Morphology

enabled dipole inversion for quantitative susceptibility mapping using structural

consistency between the magnitude image and the susceptibility map. NeuroImage

2012;59:2560-8.

33. Wu B, Li W, Avram AV, Gho S-M, Liu C. Fast and tissue-optimized mapping of

magnetic susceptibility and T2* with multi-echo and multi-shot spirals. NeuroImage

2012;59:297-305.

34. Liu T, Surapaneni K, Lou M, Cheng L, Spincemaille P, Wang Y. Cerebral

microbleeds: burden assessment by using quantitative susceptibility mapping.

Radiology 2012;262:269-78.

35. Liu S, Neelavalli J, Cheng Y-CN, Tang J, Mark Haacke E. Quantitative

susceptibility mapping of small objects using volume constraints. Magn. Reson.

Med. 2013;69:716-23.

36. Xu Y, Haacke EM. An iterative reconstruction technique for geometric distortion-

corrected segmented echo-planar imaging. Magn. Reson. Imaging 2008;26:1406-14.

37. Zwanenburg JJM, Versluis MJ, Luijten PR, Petridou N. Fast high resolution whole

brain T2* weighted imaging using echo planar imaging at 7T. NeuroImage

2011;56:1902-7.

38. Schweser F, Deistung A, Lehr BW, Reichenbach JR. Quantitative imaging of

intrinsic magnetic tissue properties using MRI signal phase: An approach to in vivo

brain iron metabolism? NeuroImage 2011;54:2789-807.

39. Neelavalli J, Cheng YN, Jiang J, Haacke EM. Removing background phase

variations in susceptibility‐weighted imaging using a fast, forward‐field calculation.

J. Magn. Reson. Imaging 2009;29:937-48.

40. Liu T, Khalidov I, de Rochefort L, Spincemaille P, Liu J, Tsiouris AJ, et al. A novel

background field removal method for MRI using projection onto dipole fields (PDF).

NMR Biomed. 2011;24:1129-36.

41. Schweser F, Deistung A, Sommer K, Reichenbach JR. Toward online reconstruction

of quantitative susceptibility maps: Superfast dipole inversion. Magn. Reson. Med.

2013; 69(6):1582-94;

42. Liu T, Wisnieff C, Lou M, Chen W, Spincemaille P, Wang Y. Nonlinear

formulation of the magnetic field to source relationship for robust quantitative

susceptibility mapping. Magn. Reson. Med. 2013;69:467-76.

43. Hammond KE, Lupo JM, Xu D, Metcalf M, Kelley DAC, Pelletier D, et al.

Development of a robust method for generating 7.0 T multichannel phase images of

the brain with application to normal volunteers and patients with neurological

diseases. NeuroImage 2008;39:1682-92.

44. Robinson S, Grabner G, Witoszynskyj S, Trattnig S. Combining phase images from

multi-channel RF coils using 3D phase offset maps derived from a dual-echo scan.

Magn. Reson. Med. 2011;65:1638-48.

45. Jenkinson M. Fast, automated, N‐dimensional phase‐unwrapping algorithm. Magn.

Reson. Med. 2003;49:193-7.


12

46. Abdul-Rahman HS, Gdeisat MA, Burton DR, Lalor MJ, Lilley F, Moore CJ. Fast

and robust three-dimensional best path phase unwrapping algorithm. Appl. Opt.

2007;46:6623-35.

47. Li W, Wu B, Liu C. Quantitative susceptibility mapping of human brain reflects

spatial variation in tissue composition. NeuroImage 2011;55:1645-56.

48. Feng W, Neelavalli J, Haacke EM. Catalytic multiecho phase unwrapping scheme

(CAMPUS) in multiecho gradient echo imaging: removing phase wraps on a voxel-

by-voxel basis. Magn. Reson. Med. 2013;70:117-26.

49. Salomir R, de Senneville BD, Moonen CT. A fast calculation method for magnetic

field inhomogeneity due to an arbitrary distribution of bulk susceptibility. Concepts

Magn. Reson. Part B Magn. Reson. Eng. 2003;19B:26-34.

50. Tang J, Liu S, Neelavalli J, Cheng YCN, Buch S, Haacke EM. Improving

susceptibility mapping using a threshold-based K-space/image domain iterative

reconstruction approach. Magn. Reson. Med. 2013;69:1396-407.

51. Schweser F, Sommer K, Deistung A, Reichenbach JR. Quantitative susceptibility

mapping for investigating subtle susceptibility variations in the human brain.

NeuroImage 2012;62:2083-100.

52. Liu T, Spincemaille P, de Rochefort L, Kressler B, Wang Y. Calculation of

susceptibility through multiple orientation sampling (COSMOS): A method for

conditioning the inverse problem from measured magnetic field map to

susceptibility source image in MRI. Magn. Reson. Med. 2009;61:196-204.

53. Shmueli K, de Zwart JA, van Gelderen P, Li T, Dodd SJ, Duyn JH. Magnetic

susceptibility mapping of brain tissue in vivo using MRI phase data. Magn. Reson.

Med. 2009;62:1510-22.


13

Chapter 2 Basic Concepts of Phase,

Gradient Echo Imaging and

Quantitative Susceptibility Mapping1

2.1 The concept of a gradient echo and phase

Gradient recalled echo (GRE) sequence is one of the most frequently used imaging

methods in MRI. To start our story, let’s first take a look at the gradient echo signal

formation mechanism. When placed in an external magnetic field, , the spins or protons

will precess about at the Larmor frequency defined as:

[2.1],

where is the gyromagnetic ratio of protons (2.68∙108∙rad∙s

-1∙T

-1). Upon the excitation by

a radial-frequency (RF) pulse, e.g., a 90o pulse, the longitudinal magnetization will be

tipped into the transverse plane. The longitudinal magnetization will gradually recover

1Most of the contents in this chapter are adapted from Haacke EM, et al. Magnetic Resonance Imaging:

Physical Principles and Sequence Design. 1st ed. Wiley-Liss; 1999, and Haacke EM, Reichenbach JR,

editors. Susceptibility Weighted Imaging in MRI: Basic Concepts and Clinical Applications. 1st ed. Wiley-

Blackwell; 2011.


14

toward the equilibrium state parallel to . This process is described using the Bloch

equations (1). Particularly, there are two important relaxation times involved in this

process, being the spin-lattice relaxation time T1 and the spin-spin relaxation time T2.

While the former relaxation time describes the regrowth rate of the longitudinal

magnetization, the latter represents the decaying rate of the magnetization in the

transverse plane. Practically speaking, due to global and local field inhomogeneities of

various origins, the spins in the transverse plane will experience extra dephasing effects

and thus the magnetization will decay much faster. This expedited decay rate is described

using the T2* time constant, which is defined as:

[2.2],

where , or

corresponds to the dephasing effects caused by field inhomogeneities.

In a 3D imaging experiment with linearly varying gradients , and

applied in three orthogonal directions, the local magnetic field becomes:

[2.3].

The spin isochromats will precess at the Larmor frequencies proportional to the local

magnetic field, and the accumulated phase induced by the linear gradient can be written

as:

∫

∫

∫

[2.4].

It is shown that, with relaxation effects neglected, the signal can be written as (1):


15

∭ [2.5],

where is the effective proton density. Let’s further define:

∫

∫

∫

[2.6],

where . Using Eqs. 2.4 and 2.6, Eq. 2.5 becomes:

( ) ∭ [2.7].

Eq. 2.7 clearly indicates that the signal is the Fourier transform of the effective proton

density, . Therefore we name the signal space defined by Eq. 2.7 as “k-space”.

With an inverse Fourier transform, the complex data of can be obtained in the

spatial domain. Magnitude and phase images can then be extracted from this complex

data.

In order to reconstruct the effective proton density, sufficient coverage of k-space is

required. This is achieved by varying the duration or amplitude of the gradients ,

and . Assuming that is the readout gradient, and are the

phase encoding and slab selection gradients, the timing and amplitudes of these gradients

can be described by the sequence diagram (2). A typical 3D gradient echo sequence

diagram is shown in Fig. 2.1.


16

Figure 2.1 Sequence diagram of a single echo 3D gradient echo sequence. GS: slab

selection/partition encoding gradient; GP: phase encoding gradient; GR: readout gradient.

The k-space trajectory can be understood from Fig. 2.2. Assuming that the duration of

both the slab selection gradient Gs and the phase encoding gradient GP is tP, the amplitude

of GP is at its maximum , at t=t1, using Eq. 2.6, the k-space data points being

sampled can be represented as:

[2.8].

This corresponds to the data points in the first row in Fig. 2.2.


17

Figure 2.2 Illustration of filling one line in k-space. The k-space trajectory as indicated

by the dashed arrows is described using Eqs.2.8 to 2.10.

Starting from t=t1, the negative lobe of the readout gradient (with amplitude -GR) will be

applied, using Eq. 2.6 again, the k-space trajectory in x direction can be written as:

[2.9].

This corresponds to moving from toward in k-

space. The positive lobe of the readout gradient is applied starting from . The k-

space trajectory in direction becomes:

[2.10].


18

When (that is when ), and an echo is formed. The duration

from the RF pulse to t3 is defined as the echo time (TE), as shown in Fig. 2.1. Starting

from t=t2, a total of Nx data points will be sampled symmetrically about , as

illustrated in Fig. 2.2. This corresponds to the coverage of to

in direction (2). The other data points in k-space are sampled similarly, by varying

Gs and Gp (hence and ) in following RF excitations. The duration between the two

RF excitations is the repetition time, TR.

Note that, the sequence shown in Fig. 2.1 is a simplified gradient echo sequence. For SWI

and QSM data acquisition, flow compensation gradients in slab select, partition encoding,

phase encoding and readout directions are usually required, in order to reduce the

dephasing effects caused by flow (1,2). The sequence diagram of a 3D gradient echo

sequence with full flow-compensation is shown in Fig. 2.3 (2). Before the excitation by

the next RF pulse, in the end of the acquisition during one RF excitation, there may be

remnant transverse magnetization, which can be destructed through spoiling. This is

achieved by keeping the gradient in the readout direction on to properly dephase the

remnant transverse magnetization (i.e., gradient spoiling), and by changing the offset

angle of the following RF pulse by 117o (i.e., RF spoiling) (2). Considering the relaxation

effects, for a particular voxel with several isochromats, the steady-state signal for the

spoiled gradient echo sequence can be written as (1,2):

[2.11],

where is the voxel spin density and .


19

Figure 2.3 Sequence diagram of a single echo 3D gradient echo sequence with flow-

compensation in all directions. GS: slab selection/partition encoding gradient; GP: phase

encoding gradient; GR: readout gradient.

However, the sequences shown in Figs. 2.1 and 2.3 are single echo gradient echo

sequences. For multi-echo gradient echo sequence, within one RF excitation, the

gradients with the same settings can be used again to sample the same line in k-space, but

with different echo time (1,3,4).


20

When there are inhomogeneities in the main magnetic field, at t=TE, with any flow

induced effects compensated, the accumulated phase for a right-handed system can be

written as (1,2):

[2.12],

where is the time-independent phase offset, related to local conductivity and

permittivity (5). The field variation is induced by inhomogeities of the main

magnetic field, susceptibility differences in the tissues in the human body and chemical-

shift. Particularly, the field variation induced by the susceptibility differences can be

predicted through a forward calculation.

2.2 Predicting field variation through forward calculation

As a basic tissue property and important source of imaging contrast, magnetic

susceptibility describes the ability of the material to get magnetized when exposed to an

external magnetic field (1,2). It is also a measure of how materials change the local

magnetic field (2). Based on their induced magnetization, the materials can be categorized

as paramagnetic, diamagnetic and ferromagnetic materials. For paramagnetic materials,

the induced magnetic moments align parallel with the external magnetic field, while for

diamagnetic materials the induced moments align anti-parallel with the external field. For

ferromagnetic materials, a magnetic field exists even without the external field (2).

When an object with susceptibility is placed in an external magnetic field

, [2.13],


21

where is the permeability and (in A/m) is the applied field (1). The actual field inside

the object can be written as:

[2.14],

where Wb/(A·m) is the permeability of vacuum, and is the induced

magnetization, which is related to the H-field through:

[2.15].

is the magnetic susceptibility. For paramagnetic materials, is positive; for

diamagnetic materials, is negative. In studies on biological tissues such as brain tissues,

the reference of susceptibility is usually taken to be the susceptibility of soft tissue or

water, with the susceptibility of water being approximately -9 ppm relative to vacuum.

Thus, being “paramagnetic” or “diamagnetic” in this thesis, essentially means that being

less diamagnetic (paramagnetic relative to water) or more diamagnetic than water (i.e.

diamagnetic relative to water) (2,6).

From Eqs. 2.14 and 2.15, assuming that ,

(

) [2.16].

A dipole field will be generated due to the induced magnetization . Assuming that the

external magnetic field is in the z direction, only the z-components of the dipole field and

are important (6–8). This z-component of the field variation can be written as (6–8):

∫

( )( )

( )

[2.17]


22

Eq. 2.17 can be written as a convolution process (9):

[2.18].

The 3D Green’s function is:

[2.19],

where is the angle subtended by a position vector, relative to the z direction in 3D

spherical coordinate system(6–8). Particularly,

.

From Eqs. 2.16 to 2.19, given the susceptibility distribution, the induced magnetic field

variation can be predicted as:

[2.20].

The convolution in Eq. 2.20 can be efficiently calculated in Fourier domain as:

[2.21],

where and represents the Fourier transform and the inverse Fourier transform,

respectively.

It can be shown that the Fourier transform of the Green’s function is (6–8):

( ) {

[2.22].

Given a susceptibility distribution, the induced field variation can be predicted using Eqs.

2.20 to 2.22. This process is referred to as the “forward calculation”.


23

2.3 Quantifying Susceptibility as an inverse problem

The field variation can be extracted from , using Eq. 2.12, and susceptibility

distribution can be calculated using through Eqs. 2.20 to 2.22. In practice,

however, susceptibility quantification is composed of several steps. The QSM data

processing procedures are illustrated in Fig. 2.4.

Figure 2.4 QSM data processing procedures. The dashed line indicates that brain masks

may not be required for phase unwrapping. The * indicates that the phase unwrapping

step can be avoided in certain algorithms and the unwrapped phase is not required for

background field removal.

Susceptibility quantification is an ill-posed inverse problem, due to the zeros in ( )

along the magic angles in the Fourier domain. This inverse problem could be solved

through either truncated k-space division (10–12) or other optimization methods (13–16).

For the former approach, a regularized inverse filter is used, defined as:


24

( ) { (

)

|

|

(

) (

)

|

|

[2.23],

and can be calculated as:

{ ( ) } [2.24].

In other approaches, is obtained as:

‖ ( )‖

[2.25],

where W is a weighting function which is related to the signal-to-noise ratio (SNR) of the

data and is the regularization term. is a parameter which controls the trade-off

between data fidelity and data regularization (13,17).

In addition, the original phase images are usually aliased. The value range of the original

phase images is from –π to π. However, the true value of phase may be outside this range

and will be wrapped back into the range of –π to π. This wrapping process can be

described as:

[2.26],

where the values of the function are integers. In order to obtain from the

phase images without discontinuities, is generated normally through phase

unwrapping, in which the function is determined.


25

Furthermore, the original phase information is contaminated by the background field

inhomogeneities induced by the air-tissue interfaces. The background field can be

removed with appropriate filtering of the data (18,19). The resulting local phase

information will be used to calculate susceptibility maps by solving the inverse problem.

More details on removing the background field and solving the inverse problem are

provided in later chapters.

2.4 Susceptibility and its relations to venous oxygen saturation

and iron content

One of the important applications of susceptibility mapping is measuring venous oxygen

saturation in vivo. Blood can be modeled using two compartments: the plasma and the red

blood cells. The susceptibility of the blood system can be written as (1,2):

( ) [2.27],

where Hct is the fractional hematocrit, defined as the volume fraction of the red blood

cells in blood. The normal value of Hct is around 0.45 for men and 0.4 for women (1,2). Y

is the oxygenation level. and are the susceptibilities of fully oxygenated and

fully deoxygenated red blood cells, respectively. Using Eq. 2.27, the susceptibility

difference between the venous blood (with 0<Y<1) and the fully oxygenated blood (Y=1)

can be written as:

[2.28],


26

where is the susceptibility difference between fully deoxygenated and fully

oxygenated red blood cells, which is approximately ppm per unit Hct (2,20,21).

Assuming that the susceptibility of fully oxygenated blood is the same as the

susceptibility of the surrounding tissue, the difference between the susceptibility of

venous blood and the surrounding tissue, , can be approximated using Eq. 2.28.

Particularly, with Hct=0.44, Y=70%, is found to be close to 0.45ppm. It is this

susceptibility difference that induces the field variation and the phase of the veins.

Apart from measuring venous oxygen saturation, various studies have been performed to

study the relation between the measured susceptibility and the iron content. Note that,

since the susceptibility maps only provide a relative measure, it is only the slope between

the measured susceptibility and iron content that can be determined. For ferritin, when the

iron content was measured by Inductively Coupled Plasma Mass Spectrometry (ICP-MS)

(22) and X-ray fluorescence (XRF) (23), the slope was found to be 1.1 ppb per iron/g

wet tissue (23). For the deep grey matter structures in cadaveric brain, however, the slope

was found to be 0.8 ppb per iron/g wet tissue, with iron measured using XRF (23). The

smaller slope found in cadaveric brain than the one found in ferritin phantom was

attributed to invisible forms of iron in the cadaveric brain and signal loss or bias caused

by the QSM data processing (22,23). Indeed, due to the singularities of the inverse kernel,

regularization is required and is known to cause under-estimation of the susceptibility.

More importantly, the quality and accuracy of QSM is largely dependent on the extraction

of local phase information and hence the efficacy of the background field removal.


27

References

1. Haacke EM, Brown RW, Thompson MR, Venkatesan R. Magnetic Resonance

Imaging: Physical Principles and Sequence Design. 1st ed. Wiley-Liss; 1999.



3. Bernstein MA, King KF, Zhou XJ, editors. Chapter 11 - Signal Acquisition and k-

Space Sampling (Internet). In: Handbook of MRI Pulse Sequences. Burlington:

Academic Press; 2004 (cited 2014 Jan 22). page 367–442.Available from:

http://www.sciencedirect.com/science/article/pii/B9780120928613500170

4. Ye Y, Hu J, Wu D, Haacke EM. Noncontrast-enhanced magnetic resonance

angiography and venography imaging with enhanced angiography. J. Magn. Reson.

Imaging 2013;38:1539–48.

5. Haacke EM, Petropoulos LS, Nilges EW, Wu DH. Extraction of conductivity and

permittivity using magnetic resonance imaging. Phys. Med. Biol. 1991;36:723.

6. Marques J p., Bowtell R. Application of a Fourier-based method for rapid

calculation of field inhomogeneity due to spatial variation of magnetic susceptibility.

Concepts Magn. Reson. Part B Magn. Reson. Eng. 2005;25B:65–78.

7. Koch KM, Papademetris X, Rothman DL, de Graaf RA. Rapid calculations of

susceptibility-induced magnetostatic field perturbations for in vivo magnetic

resonance. Phys. Med. Biol. 2006;51:6381–402.

8. Cheng Y-CN, Neelavalli J, Haacke EM. Limitations of calculating field distributions

and magnetic susceptibilities in MRI using a Fourier based method. Phys. Med. Biol.

2009;54:1169–89.

9. Deville G, Bernier M, Delrieux JM. NMR multiple echoes observed in solid 3He.

Phys. Rev. B 1979;19:5666–88.



Med. 2009;62:1510–22.



2010;32:663–76.


of quantitative susceptibility maps: superfast dipole inversion. Magn. Reson. Med.

2013;69:1582–94.




2012;59:2560–8.



NeuroImage 2012;62:2083–100.

15. Wu B, Li W, Guidon A, Liu C. Whole brain susceptibility mapping using

compressed sensing. Magn. Reson. Med. 2012;67:137–47.


28



reconstruction approach. Magn. Reson. Med. 2013;69:1396–407.

17. De Rochefort L, Brown R, Prince MR, Wang Y. Quantitative MR susceptibility

mapping using piece‐wise constant regularized inversion of the magnetic field.

Magn. Reson. Med. 2008;60:1003–9.



NMR Biomed. 2011;24:1129–36.



brain iron metabolism? NeuroImage 2011;54:2789–807.

20. He X, Yablonskiy DA. Biophysical mechanisms of phase contrast in gradient echo

MRI. Proc. Natl. Acad. Sci. U. S. A. 2009;106:13558–63.

21. Jain V, Langham MC, Wehrli FW. MRI estimation of global brain oxygen

consumption rate. J. Cereb. Blood Flow Metab. 2010;30:1598–607.

22. Langkammer C, Schweser F, Krebs N, Deistung A, Goessler W, Scheurer E, et al.

Quantitative susceptibility mapping (QSM) as a means to measure brain iron? A

post mortem validation study. NeuroImage 2012;62:1593–9.

23. Zheng W, Nichol H, Liu S, Cheng YCN, Haacke EM. Measuring iron in the brain

using quantitative susceptibility mapping and X-ray fluorescence imaging.

NeuroImage 2013;78:68–74.


29

Chapter 3 Background Field Removal

3.1 The background field

Phase information is proportional to the product of field variation and echo time (TE)

(1,2). For a right-handed system, the phase is related to the total field variation

as:

[3.1],

where is the phase offset at TE=0. As mentioned in the earlier chapter, the original

phase images are usually aliased and phase unwrapping should be performed in order to

obtain using Eq. 3.1. The field variation can be considered as a

combination of two parts (3–6): the background field and the local field :

[3.2].

In quantitative susceptibility mapping, it is the local field variation that is of

interest. The background field is induced by the main magnetic field inhomogeneity and

the air-tissue interfaces which have a susceptibility difference of approximately 9.4 ppm

relative the brain tissue (4). This is much higher than the susceptibility differences

(<1ppm) of the local tissues inside the brain (4–7). At most realistic echo times, this large


30

field variation leads to phase aliasing, further complicating the problem of background

field removal. Another complication occurs in the presence of a local background

gradient which will cause a shift of the echo center in k-space which, in turn, induces a

linear phase shift in the image domain (1). This can be easily understood through the

Fourier shift theorem. For example, if the center of the echo is shifted in by , a

linear phase component with gradient will be created in the image domain.

This linear phase component can be easily removed by detecting the position of the signal

center (peak value) in k-space and shifting the k-space data such that the signal center

occurs at the actual center of k-space (1).

To remove the background field, various algorithms have been proposed. There are

mainly four types of methods: geometry dependent artefact correction (GDAC) (4),

(homodyne) high-pass filtering (8–10), spherical mean value filtering (3,6,7) and dipole

field fitting (5), all with different assumptions of the background field.

If the geometries and susceptibility distributions of the air-tissue interfaces were known,

the field variation induced by the air-tissue interfaces could be predicted using the fast

forward field calculation (4). Assuming that the susceptibility differences of these

air/bone-tissue interfaces are for the ith

interface, with i ranging from 1 to n, the

induced field variation can be calculated as:

[3.3].

The geometry of the sinuses can be extracted from the T1 weighted magnitude image, and

the susceptibilities of different sinuses can be found through least squares fitting using


31

phase images without wraps (4). This method has been shown to effectively reduce the

background field. However, obtaining the exact geometries and susceptibility

distributions of the air/bone-tissue interfaces, such as the air sinuses in the head, is quite

challenging. This is partly solved by assuming constant susceptibilities inside the sinuses.

Recently, a method of using short TE’s data was proposed, which produce susceptibility

maps for the air sinuses and bone structures (11). This can potentially solve the remnant

problems of geometry dependent artefact correction.

In the other background field removal algorithms, the background field is reduced, or

separated from the local field, without knowing the exact geometries or susceptibility

distributions of the air-tissue interfaces. These filtering methods are discussed in

following sections. Note that, since phase is a linear function of field variation, as shown

in Eq. 3.1, the filters discussed in this thesis are operated directly on the (unwrapped)

phase images rather than the field maps.

3.2 Homodyne high-pass filter

3.2.1 Theory

Homodyne high-pass filtering is the traditional way for background field suppression (10).

It is widely used in susceptibility weighted imaging (SWI), due to its robustness and

effectiveness. It makes use of the fact that the background field has relatively low spatial

frequency. Consequently, the background field can be reduced through high-pass filtering.

In homodyne high-pass filtering, low-pass filtered data are created by applying Hanning

window to the central part of k-space (2,12).


32

[3.4],

Where is the original complex data and is a zero-filled 2D Hanning

window defined as:

( )

{

[ (

(

)

)] [ (

(

)

)]

[3.5].

Then the low-pass filtered data, , are complex divided into the original complex data ,

to produce high-pass filtered complex data . The high-pass filtered phase images, ,

are generated as:

( ) ( ) [3.6],

where “/” represents pixel-wise division. The whole process of homodyne high-pass

filtering is shown in Fig. 3.1.

One advantage of homodyne high-pass filtering is that no phase unwrapping is required.

This is due to the use of the complex data in the high-pass filtering. On the other hand, the

phase images can also be unwrapped first and then a high-pass filter can be applied in the

image domain, e.g., applying a Gaussian or boxcar window function to the unwrapped

phase images (2). Both the homodyne high-pass filtering and image domain high-pass

filtering have edge artefacts, since there is low/no reliable signal outside the brain.


33

Figure 3.1 Illustration of the processing steps in homodyne high-pass filtering.

3.2.2 Effects of object orientation in homodyne high-pass filtering

Homodyne high-pass filtering has been used successfully in SWI, especially for

venography, in which the objects of interest are the veins. However, high-pass filtering

also leads to signal loss, especially for objects with relatively large size, such as those

basal ganglia structures. This will cause underestimation of the susceptibility of those

basal ganglia structures (2,13). Apart from the filter size, the accuracy of homodyne high-

pass filtering is also dependent on the direction along which the high-pass filter is applied,

since the homodyne high-pass filter is typically applied in 2D. This may lead to different

levels of signal loss to structures with different orientation (14).

In order to study the effect of homodyne high-pass filtering, phase images of cylinders

and spheres with different radii were simulated in a 256×256×256 matrix, with B=3T,

TE=20ms. The cylinders were used as a surrogate for veins, and spheres for microbleeds.


34

The susceptibilities inside the cylinders and spheres were set to 0.45ppm and 1ppm,

respectively. The radii of the cross section of the cylinders range from 1 to 32 pixels. The

radii of the spheres are in the same range. For the cylinders, the long axis was set to be 30,

45, 60, and 90 degrees with respect to the main field direction. Homodyne high-pass filter

with different sizes were applied to the cylinders in 90o with respect to the main field

direction, in order to show the effects of filter size. Next, to show the effects of

orientation of the object, homodyne high-pass filter with size 32×32 were applied in axial,

coronal and sagittal views of the phase images of cylinders and spheres. Susceptibility

maps were generated using truncated k-space division (15), with a k-space threshold of

0.1, using both the original and filtered phase images. Mean susceptibility values were

measured for the regions inside the cylinders and spheres.

The relative errors in susceptibilities as a function of object size for different filter sizes,

for cylinders perpendicular to the main magnetic field are shown in Fig.3.2.a. As

expected, the larger the filter size, the greater the signal loss and hence the greater the

underestimation in the estimated susceptibility values.

The relative errors plotted in Fig. 3.2.b to 3.2.f are dependent on the object and filtering

orientations. For cylinders, the least error in the estimated susceptibilities was obtained

when the HP filter was applied in the direction which is perpendicular to the long axis of

the cylinder. For example, in Fig. 3.2.b, the cylinders were aligned along the x direction.

When the HP filter is applied to the view perpendicular to the long axis of the cylinders

(the y-z view of the dataset), the errors were smaller than HP filtering in other views. For

the 30o, 45

o and 60

o cases, none of the slice orientations were perpendicular to the long


35

axis of the cylinders. Hence varying error profiles were observed for each case, as shown

in Fig. 3.2.c to 3.2.e. For these cases, the least errors in the estimated susceptibilities were

obtained when the HP filter was applied in the orientation which is closest to be

perpendicular to the long axis of the cylinder. This can be seen from Fig. 3.2.c and 3.2.e.

In Fig. 3.2.c, the least error is obtained when the HP filter is applied to the sagittal view,

while in Fig. 3.2.d to the coronal view.

When the angle between the long axis of the cylinders and the field direction is 45o, as

would be expected, both axial and coronal views give the same error profile (Fig. 3.2.d).

This relation between errors in susceptibility estimation and HP filtering orientation was

further confirmed by Fig. 3.2.f, in which the errors for spheres were independent to the

filtering orientation.

2D homodyne high-pass filtering, which is currently used in SWI data processing, leads

to orientation-dependent errors to the estimated susceptibilities of cylinders.

Consequently, when the homodyne high-pass filtering was used in the quantification of

susceptibilities of veins, the orientation of the filtering with respect to the veins should be

considered. This orientation dependence can be potentially removed by using a 3D

homodyne high-pass filter. For data collected which were already filtered using a 2D

homodyne high-pass filter, an additional homodyne high-pass can be applied in the

through-plane direction, to form an effective 3D high-pass filtering.


36

Figure 3.2 Relative error in estimated susceptibilities induced by high-pass filtering in

different orientations. The relative errors induced by homodyne high-pass filtering with

different sizes, for cylinders in the 90o case, are shown in a. b to e show the effects of the

orientation of high-pass filtering for cylinders in the 90o case (b), in the 30

o case (c), in

the 45o case (d), and in the 60

o case (e). Relative errors for the spheres are shown in f.


37

3.3 Variable high-pass filter (VHP)

3.3.1 Theory

It is known that homodyne high-pass filtering leads to signal loss and underestimation of

the susceptibilities of structures with relatively large sizes, such as the basal ganglia

structures. In fact, variable filter sizes can be used for different regions of the brain (16).

The main idea is to apply mild filtering to the central part of the brain, where most of the

relatively large structures are located, and to apply stronger filtering to the periphery of

the brain, especially to the regions close to the air sinuses, where the background field has

relatively higher spatial frequency.

A major advance in this area of research on background field removal occurred, when it

was recognized that the background field can be considered as a harmonic function inside

the brain (3,6). Using the mean value property of harmonic functions,

[3.7],

where “s” is a normalized sphere with uniform values and the sum of all the values of the

pixels inside the sphere equals 1. “*” represents the convolution operation. From Eqs. 3.2

and 3.7,

[3.8].

Eq. 3.8 can also be viewed as a low-pass filtering process using a spherical symmetric

window function, s. Next, the background field could be removed by subtracting the low-

pass filtered field map from the original field map:


38

[3.9],

where is the delta function.

However, using the above equation, the regions within a distance of R to the edge of the

brain do not have correct convolution results (6,7), where R is the radius of the spherical

kernel. Those regions will be set to 0 using a 3D eroded mask. To reduce the area of the

lost regions, we applied spherical kernels with variable size, similar to the strategy used in

(7). Specifically, for the periphery region of the brain, the size of the kernel at certain

pixel is chosen to be the distance from that pixel to the nearest edge pixel of the brain.

While for the central region of the brain, a large kernel with constant size was applied.

This is essentially a high-pass filtering process, and is referred to as variable high-pass

filtering (VHP) in this thesis.

The procedures of variable high-pass filtering are as follows:

1. Preprocessing: generate through phase unwrapping; obtain the brain masks

defining the region inside the brain.

2. Spherical averaging: apply the normalized spherical kernels with different radii to

. will be produced by combining the results generated using different

spherical kernels.

3. High-pass filtering: will be generated as .


39

3.3.2 Evaluating the accuracies of VHP using a simulated brain model

Brain model data simulation

In order to evaluate the accuracy of VHP, phase images of a 3D brain model were

simulated using the forward field calculation (17,18), at B0=3T, TE=10ms. The

background field variation induced by these air-tissue interfaces were introduced by

setting the susceptibility of the air sinuses to 9.4 ppm. Different susceptibility values were

set to different brain structures. The phase induced by the brain structures (the local

phase) and the phase induced by the sinuses (the background phase) were

calculated independently. Then the phase images of the brain model ( ) were

generated by taking the sum of these two components. The magnitude images were

assumed to be constant. White Gaussian noise was added to real and imaginary parts of

the complex data, and the SNR in the final magnitude images was 10:1. This is equivalent

to Gaussian noise with standard deviation 0.1 radians in phase images (1,19). The brain

masks were generated which covers all the structures inside the brain. A reference region

was obtained by eroding the brain masks 16 pixels. This reference region was used to

calculate the root-mean-square-error (RMSE) in the processed phase images and in the

generated susceptibility maps. The simulated phase images and the binary masks are

shown in Fig. 3.3.

The simulated phase images with background phase component and random noise were

processed using VHP with largest spherical kernel radius 32 pixels and smallest spherical

kernel radius 1 pixel. The processed phase images were compared with the simulated


40

phase of the structures inside the brain ( ), without any background phase or random

noise. Root-mean-square-error (RMSE) was calculated for the reference region as:

√ ( ) [3.10],

where is the total number of pixels in the reference region.

Figure 3.3 a). Simulated phase image with background phase but without random noise.

b). Simulated phase with background phase and random noise. c). The local phase

information without random noise. This is used as the true answer to evaluate the

accuracies of the processed phase images. d). The reference region used for calculating

the RMSEs in the processed phase images. e). The susceptibility map in axial view. f).

The phase image corresponds to e.


41

In order to evaluate the accuracy of susceptibility quantification using VHP processed

phase images, susceptibility maps were generated using the truncated k-space division

approach (15). The mean susceptibility values were measured for a total of 9 structures,

as listed in Table 3.3. The relative error in the susceptibility quantification of each

structure was calculated as: , where is the measured mean

susceptibility value and is the true susceptibility value of certain structure. However,

this error is a combination of both the errors caused by background field removal and the

errors by the truncated k-space division susceptibility mapping algorithm. To analyze the

error purely caused by background field removal, reference susceptibility maps were

generated with the same algorithm using the phase images with ideal background field

removal, i.e., the local brain structures induced phase with random noise.

Simulated data results

As expected, both the RMSEs of the processed phase image and the susceptibility maps

reduce as the radius of the spherical kernel increases, as shown in Fig. 3.4. This can be

understood as a reduction of signal loss related directly to an increase in the size of the

spherical kernel in image domain. The results shown in Fig. 3.4 correspond to the global

errors.


42

Figure 3.4 Overall RMSEs of the processed phase images (a) and susceptibility maps (b)

generated using VHP.

The relative errors in the susceptibility maps for different structures are plotted in Fig. 3.5.

For small structures such as veins, as the radius of the spherical kernel increases, the

relative error reduces much faster than for other bigger structures such as the globus

pallidus. Compared with the error in the susceptibilities estimated from the original phase

(as indicated by the dashed lines in Fig. 3.5), when variable high-pass filtering was

applied, the additional under-estimation could be around 20% to 30% for big structures

such as putamen.


43

Figure 3.5 Relative errors in the estimated susceptibilities of different structures using

phase images processed by VHP. a) Veins. b) Globus pallidus. c) Putamen. d) Caudate.

The dashed lines indicate the errors in estimated susceptibilities for different structures

using the phase images with ideal background field removal.

For the processed phase images, the larger the radius of the spherical kernel, the smaller

the error caused by spherical filtering. When spherical kernels with different radii were

applied to different regions, the error in the estimated susceptibility of a certain structure

can be different due to the different locations of the structures. The most affected

structures are those which extend significantly spatially, e.g., putamen. Compared with


44

traditional high-pass filtering with a single filter size, this variable high-pass filter helps to

preserve the low spatial frequency signal of the big structures while providing accurate

phase information of the veins.

3.4 Sophisticated Harmonic Artifact Reduction for Phase data

(SHARP)

3.4.1 Theory

The same set of equations used in VHP, Eqs. 3.7 to 3.9, are used in SHARP to remove the

background field, since SHARP is also based on the spherical mean value property of the

background field. Nonetheless, as can be seen from Eq. 3.9, the obtained local field

variation is essentially high-pass filtered. Thus, the last step in SHARP is to solve Eq. 3.9

for the local field variation as an inverse problem (6). This deconvolution process

is done typically using a truncated version of the filter in Fourier domain, similar to the

truncated k-space division method for QSM described in Chapters 2 and 5. Again, the

pixels close to the edges will not have the correct convolution result, and will have to be

removed using an eroded brain mask M. The last step in SHARP can be written as:

[3.11],

where the regularized inverse filter is defined as:

( ) {

[3.12].


45

Another way is to solve this inverse problem as an optimization problem with a priori

information incorporated (20).

3.4.2 The influences of processing parameters on the accuracy of SHARP

The accuracy of SHARP is dependent on both the radius of the spherical kernel and the

deconvolution process. In this study, the deconvolution was done using the regularized

inverse filter shown in Eq. 3.11. In order to evaluate the effects of both the radius of the

spherical kernel and the regularization threshold th, and to determine the optimal

processing parameters in SHARP, SHARP with different spherical kernel sizes and

different threshold values were tested using the simulated phase images of the 3D brain

model. The same 3D brain model which was used to evaluate the accuracy of VHP was

also used in this section. Particularly, the radii of the spherical kernel ranged from 1 pixel

to 16 pixels with step size 1 pixel, and the values of th ranged from 0.005 to 0.1 with step

size 0.005. Again, RMSEs in the processed phase images and the susceptibility maps

were used to evaluate the accuracy of SHARP. The relative errors in the measured

susceptibilities for different structures were used to assess the effects of SHARP on

susceptibility quantification of different structures.

Fig. 3.6.a shows the overall RMSE in the phase images processed using SHARP. The

overall RMSE in the susceptibility map, generated using the SHARP processed phase

images is shown in Fig. 3.6.b. Generally speaking, spherical kernels with larger radii lead

to smaller errors. For individual structures, the optimal spherical kernel radius and

regularization threshold may vary a lot, as shown in Figs 3.7 and 3.8.


46

Figure 3.6 a) Overall RMSEs for different radii of the spherical kernel and different

values of th in SHARP. b) Overall RMSE in susceptibility quantification for different

kernel sizes and different thresholds in SHARP.

The relative errors in the measured susceptibilities for different structures are illustrated

in Fig. 3.7. Large relative error in the estimated susceptibility was observed for thalamus,

where the error was higher than 50%. This could be caused by the low susceptibility

value of the thalamus and by the streaking artefacts from the veins nearby. For the other

structures, in order to maintain a low level of error, the proper choice of the kernel size

depends on the sizes of the structures. For veins, spherical kernel with radius larger than 3

pixels leads to reasonable accuracy. But for larger structures, such as the putamen, the

radius of the spherical kernel has to be at least 6 pixels.


47

Figure 3.7 Relative errors in measured susceptibilities using different parameters in

SHARP for different structures.

For different structures, the minimal relative errors in the estimated susceptibility values

as functions of the radius of the spherical kernel are plotted in Fig. 3.8. The minimal

errors in susceptibility estimates decreases as the radius of the spherical kernel increases

for all the structures, except for substantia nigra, crus cerebri and subthalamic nucleus.


48

Figure 3.8 The minimal RMSEs for different structures at different radii of the spherical

kernel in SHARP.

Considering the relatively higher signal loss at larger kernel sizes, a reasonable choice for

the size of the spherical kernel in SHARP should be around 6-10 pixels (16). For the

simulated data used in this study, the regularization threshold should be kept as small as

possible (th<=0.05). However, for in vivo data, a higher threshold may be desired, in

order to suppress non-harmonic phase components (6), e.g., the phase offset term .

Examples of the phase images processed with SHARP are available in the next section.


49

3.5 Comparison of different background field removal

algorithms

In addition to homodyne high-pass filtering, VHP, and SHARP, another popular

background field removal algorithm is Projection onto the Dipole Field (PDF). In PDF,

the background field inside the brain is assumed to be caused by the point-dipole sources

outside the brain (5). And the background susceptibility distribution is estimated as:

| |

[3.13].

Once is known, the background field can be calculated using the forward

calculation and subsequently removed. However, the estimated does not reflect

the actual background susceptibility distribution, but is only used for the purpose of

removing the background field (5). Moreover, an additional high pass filter using

spherical kernel is usually applied to the PDF processed phase images to further reduce

the low spatial frequency phase artefacts.

3.5.1 Simulations and in vivo data studies

In this section, different background field removal algorithms are compared using the 3D

brain model and in vivo data. The data of the 3D brain model were created in the same

way as in the previous section. For the in vivo data, 5 datasets were collected on the same

healthy volunteer using the parameters shown in Table 3.1. Except for the last dataset,

which was collected using a double-echo sequence (only the longer TE’s data were used

here), all the other datasets were collected using single echo gradient echo sequences. The

original phase images of the in vivo data were first unwrapped using a path-following 3D


50

phase unwrapping algorithm (21). Binary brain masks were generated using Brain

Extraction Tool (BET) in FSL (22). Other processing parameters used in different

algorithms are listed in Table 3.2. For homodyne high-pass filtering, only the 64x64 filter

was applied to the in vivo data, in order to avoid any remnant aliasing near the air-tissue

interfaces. For both the brain model and in vivo data, susceptibility maps were generated

using truncated k-space division with threshold 0.1, using the phase images processed

with different algorithms. This algorithm was chosen, primarily because of its simplicity

and well understood systematic error. The susceptibilities were measured for 9 structures

in the brain model, as shown in Table 3.3. For the in vivo data, the mean value of the

susceptibilities were measured for a total of 7 structures, including veins, globus pallidus

(GP), putamen (PUT), caudate nucleus (CN), substantia nigra (SN), red nucleus (RN) and

thalamus (THA) using manually drawn ROIs. One-way ANOVA was used to test the

differences seen in the measured susceptibilities obtained using different phase processing

algorithms, for each structure. p<0.05 was considered as significant.

3.5.2 Simulated data results

For simulated data, VHP, SHARP and PDF led to similar level of errors in the estimated

susceptibilities for different structures, as shown in Table 3.3. For basal ganglia

structures, the truncated k-space division algorithm for susceptibility mapping will cause

an underestimation around 10-20%. The VHP, SHARP and PDF processed phase images

lead to an additional 20% error for basal ganglia structures. For other structures such as

veins, the underestimation was smaller. The most severe error was seen in the results for


51

thalamus, for which the underestimations in the measured susceptibilities were large for

all three phase processing methods. Even using the phase images with ideal background

field removal in susceptibility quantification, the error was still large. This suggests that

this error is largely caused by the truncated k-space division algorithm itself.

Table 3.1 Imaging parameters for the in vivo data.

Datasets 1 2 3 4 5

TE (in ms) 14.3 17.3 12.8 19.2 17.6

TR (in ms) 26 26 34 34 30

BW (in Hz/px) 121 121 130 130 425

Voxel Size (in mm3) 0.5×0.5×

0.5

0.5×0.5×

0.5

0.6×0.6×

1.2

0.6×0.6×

1.2

0.6×0.6×

1.2

Matrix Size 512×368

×256

512×368

×196

512×336

×112

512×336

×112

512×368

×144

Flip Angle (in degrees) 15 15 15 15 15

Table 3.2 Data processing parameters in different algorithms.

Simulated data In vivo data

Homodyne

high-pass filter

k-space window sizes 32x32 and

64x64

k-space window sizes 64x64

VHP Spherical kernel radii: 32 px

(largest), 1 px (smallest)

Spherical kernel radii: 32 px

(largest), 1 px (smallest)

SHARP Spherical kernel radius: 8 px,

Regularization thresholds: 0.02

and 0.05

Spherical kernel radius: 8 px,

Regularization threshold: 0.05

PDF Convergence tolerance: 10-3

Convergence tolerance: 10-2

Spherical kernel radius: 32px


52

Table 3.3 The estimated susceptibilities (mean ± std. in ppm) for different structures in

the brain model using different phase processing methods. “Original Phase” represents

using the simulated phase images without any background field.

Original

Phase

HP32 HP64 VHP SHARP

th=0.02

SHARP

th=0.05

PDF

Veins 0.45 0.41

±0.07

0.36

±0.09

0.33

±0.09

0.40

±0.07

0.40

±0.07

0.40

±0.07

0.40

±0.07

Thalamus 0.01 0±0.06 -0.02

±0.06

-0.01

±0.06

0

±0.06

0

±0.06

0

±0.06

0

±0.06

Red

Nucleus

0.13 0.12

±0.06

0.07

±0.06

0.07

±0.06

0.11

±0.06

0.11

±0.06

0.11

±0.06

0.11

±0.06

Substantia

Nigra

0.16 0.15

±0.06

0.09

±0.07

0.09

±0.07

0.13

±0.06

0.14

±0.06

0.14

±0.06

0.14

±0.06

Subthalamic

Nucleus

0.20 0.19

±0.06

0.15

±0.07

0.14

±0.06

0.18

±0.06

0.18

±0.06

0.18

±0.06

0.19

±0.06

Crus

Cerebri

-0.03 -0.03

±0.06

-0.04

±0.07

-0.03

±0.06

-0.04

±0.06

-0.04

±0.06

-0.04

±0.06

-0.03

±0.06

Caudate 0.06 0.05

±0.06

0.03

±0.07

0.02

±0.06

0.04

±0.06

0.04

±0.06

0.04

±0.06

0.04

±0.06

Putamen 0.09 0.07

±0.06

0.02

±0.08

0.01

±0.07

0.05

±0.06

0.06

±0.06

0.05

±0.06

0.06

±0.06

Globus

Pallidus

0.18 0.15

±0.06

0.06

±0.07

0.05

±0.07

0.11

±0.06

0.14

±0.06

0.12

±0.06

0.13

±0.06


53

On the other hand, the homodyne high-pass filters caused much larger error, especially

for bigger structures such as those basal ganglia structures. Using homodyne high-pass

filter with window size 32×32 caused smaller error than using a window size 64×64, as

expected.

Moreover, the susceptibilities obtained using the simulated phase images without

background field are plotted against the susceptibilities obtained using the phase images

processed with different algorithms, as shown in Fig. 3.9. This figure reflects the effects

of different phase processing methods on susceptibility mapping. Particularly, if no

additional errors are caused by the phase processing methods, the susceptibilities

estimated using the processed phase images should be almost the same as those estimated

using the original phase images. This can be indicated by the slopes and the correlation

coefficients of the linear regressions shown in Fig. 3.9. The smaller the slope, the more

underestimation in susceptibilities the filter causes. The smaller the correlation coefficient,

the more uncertainty the filter induces. The homodyne high-pass filtering leads to more

underestimation in the estimated susceptibilities than the other phase processing methods,

and 32x32 homodyne high-pass filter causes less underestimation than 64x64 high-pass

filter does. Using the phase images processed by VHP, SHARP and PDF, the estimated

susceptibilities are almost the same as those obtained using the original unfiltered phase

images.


54

Figure 3.9 Susceptibility estimated using the original phase vs. the susceptibility

estimated using different phase processing methods: a) Homodyne HP32×32, b)

Homodyne HP64×64, c) SHARP (radius=8px, th=0.02), d) SHARP (radius=8px,

th=0.05), e) Variable HP, and f) PDF.


55

3.5.3 In vivo data results

The phase images in Dataset 1 processed using different algorithms are shown in Fig.

3.10. Edge artefacts were observed in the homodyne high-pass filter processed phase

images. The edge artefact was propagated into the susceptibility maps, as can be seen

from Fig. 3.11.a and 3.11.e. The VHP processed phase images (Fig. 3.11.b and 3.11.f)

show good agreement with the SHARP processed phase images. However, the regions

close to the edge of the brains have signal loss due to the strong high-pass filter that is

applied to these regions. The SHARP processed phase images have severe error near the

superior sagittal sinus (SSS), as shown in Fig. 3.11.c. This is due to the partial removal of

the SSS by the brain mask. Even though a mild high-pass filter was applied, a slowly

varying remnant background profile was still observed in the PDF processed phase

images, as can be seen from Fig. 3.10.d and 3.10.h. A relatively large error was found for

the structures close to the edge of the brain, using PDF processed phase images.

Using different background field removal algorithms, the measured susceptibilities for

different structures in the in vivo data are compared in Fig. 3.12. Compared with other

algorithms, homodyne high-pass filter caused more under-estimation in the measured

susceptibilities for all the structures (p<<0.05), except for the caudate nucleus (p=0.8717).

For homodyne high-pass filtering, the large variation in the measured susceptibility for

the same structure across different datasets is partly caused by the remnant phase aliasing

artefacts in the high-pass filtered phase images. VHP, SHARP, and PDF led to almost the

same susceptibility estimates for all the structures, except for the thalamus (p=0.0182).


56

Figure 3.10 Phase images processed with different algorithms in Dataset 1. a and e.

64x64 Homodyne high-pass filter. b and f: VHP. c and g: SHARP. d and h: PDF.

Figure 3.11 Susceptibility maps generated with different algorithms in Dataset 1. a and e.

64×64 Homodyne high-pass filter. b and f: VHP. c and g: SHARP. d and h: PDF.


57

Figure 3.12 The means and standard deviations of the measured susceptibility values for

different structures in different in vivo datasets. The error bars represent the standard

deviations of the measured susceptibilities in different datasets. GP: globus pallidus,

PUT: putamen, CN: caudate nucleus, SN: substantia nigra, RN: red nucleus, and THA:

thalamus. The asterisks indicate Tukey’s HSD significance: * p<0.05, ** p<0.01, ***

p<0.001.

3.5.4 Discussion and conclusions

Clearly, the variable high-pass filtering (VHP) leads to less underestimation in the

susceptibility estimation, compared with the traditional homodyne high-pass filtering. In

Veins GP PUT CN SN RN THA-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4S

usce

ptib

ility /p

pm

HP64

VHP

SHARP

PDF

**

***

**

*** **

*

*

***

*


58

VHP, the spherical kernel is used to generate a low-pass filtered phase image (the

averaged phase image) which is close to the background phase. As can be seen from the

results, variable high-pass filtering leads to similar results as SHARP. The only difference

between VHP and SHARP is that, no additional deconvolution is required in VHP. This

helps to avoid any amplification of noise due to the deconvolution. In addition, there is no

need to select any regularization parameter for deconvolution, as it is done in SHARP.

It can be concluded that the error in SHARP processed phase images reduces as the size

of the spherical kernel increases. This is mainly due to the deconvolution process, as

larger spherical kernel leads to less amplification of any remnant background field.

However, as the size of the spherical kernel increases, more of the brain region will be

affected by edge artefacts. The erosion of the brain mask leads to signal loss and is a

particular limitation of SHARP. One possible solution is to extend the field continuously

to the regions outside of the brain (16, 23).

Both SHARP and VHP processed phase images lead to small errors in the estimated

susceptibilities for most structures, except for the thalamus. This might be due to the low

susceptibility and the large size of the thalamus. For veins and basal ganglia structures,

the error is relatively more stable.

PDF processed phase images are also affected by edge artefacts. This large error close to

the edge was explained to be caused by the violation of the assumption of PDF, that is,

the field induced by the dipole sources located within the brain is orthogonal to the field

induced by the dipole sources outside the brain. Further, a remnant slowly varying


59

background phase component was observed for the PDF processed phase images, for the

in vivo data. This might be caused by the phase offset, , which could be removed when

data were collected with a multi-echo gradient echo sequence.

Generally speaking, if the original phase images are unwrapped properly, SHARP is

found to be the most efficient background field removal algorithm among all the available

algorithms. However, if a region with noisy phase values were included in the region of

interest, the phase unwrapping may fail (2) and accuracy of SHARP processed phase

images could be deteriorated by the deconvolution step (24). For instance, when there is

cusp artefact/open-ended fringe lines in the phase images, most phase unwrapping

algorithm will fail. In such cases, it would be better to simply use the variable high-pass

filter which does not perform any deconvolution, in order to avoid any amplification of

the artefacts coming from the noisy region included in the region of interest.


60

References





3. Li L. Magnetic susceptibility quantification for arbitrarily shaped objects in

inhomogeneous fields. Magn. Reson. Med. 2001;46:907–16.

4. Neelavalli J, Cheng YCN, Jiang J, Haacke EM. Removing background phase

variations in susceptibility-weighted imaging using a fast, forward-field calculation.

J. Magn. Reson. Imaging 2009;29:937–48.



NMR Biomed. 2011;24:1129–36.





spatial variation in tissue composition. NeuroImage 2011;55:1645–56.

8. Reichenbach JR, Venkatesan R, Schillinger DJ, Kido DK, Haacke EM. Small

vessels in the human brain: MR venography with deoxyhemoglobin as an intrinsic

contrast agent. Radiology 1997;204:272–7.

9. Reichenbach JR, Essig M, Haacke EM, Lee BC, Przetak C, Kaiser WA, et al. High-

resolution venography of the brain using magnetic resonance imaging. MAGMA

1998;6:62–9.

10. Haacke EM, Xu Y, Cheng YN, Reichenbach JR. Susceptibility weighted imaging

(SWI). Magn. Reson. Med. 2004;52:612–8.

11. Buch S, Ye Y, Cheng Y, Neelavalli J, Haacke EM. Susceptibility mapping of air,

bone and calcium in the head using short echo times. Magn. Reson. Med. 2013;

12. Haacke EM, Mittal S, Wu Z, Neelavalli J, Cheng Y-CN. Susceptibility-Weighted

Imaging: Technical Aspects and Clinical Applications, Part 1. Am. J. Neuroradiol.

2009;30:19–30.




14. Neelavalli J, Liu S, Cheng Y, Haacke EM, Kou Z. Effect of Orientation of 2D Phase

High-Pass Filter on Susceptibility Mapping of Veins and Microbleeds. Proc. 19th

Annu. Meet. ISMRM Montr. Quebec Canada. 2011:4517.



2010;32:663–76.

16. Li W, Avram AV, Wu B, Xiao X, Liu C. Integrated Laplacian-based phase

unwrapping and background phase removal for quantitative susceptibility mapping.

NMR Biomed. 2013; DOI: 10.1002/nbm.3056


61



2009;54:1169–89.




19. Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn. Reson.

Med. 1995;34:910–4.

20. Sun H, Wilman AH. Background field removal using spherical mean value filtering

and Tikhonov regularization. Magn. Reson. Med. 2013; DOI: 10.1002/mrm.24765



2007;46:6623–35.

22. Smith SM. Fast robust automated brain extraction. Hum. Brain Mapp. 2002;17:143–

55.

23. Topfer, R., Schweser, F., Deistung, A., Reichenbach, J. R. and Wilman, A. H. (2014),

SHARP edges: Recovering cortical phase contrast through harmonic extension.

Magn Reson Med. doi: 10.1002/mrm.25148



2013;69:1582–94.


62

Chapter 4 Fast and Robust

Background Field Removal using

Double-echo Data

4.1 Introduction

Quantitative susceptibility mapping (QSM) has great potential in elucidating tissue

properties and providing functional information in vivo. It has already found various

applications, such as the quantification of cerebral iron content (1,2), venous oxygen

saturation (3,4), and monitoring the changes of cerebral lesions and microbleeds (5,6). To

extract the susceptibility distribution, normally the phase images from a gradient echo

sequence are required. However, the phase images are contaminated by background field

inhomogeneities. The background field induced phase component is referred to as the

background phase in this study. Various algorithms have been proposed to remove the

background phase and to extract the local phase which contains the field variation induced

by local susceptibility changes (7-11). Most of these approaches require phase

unwrapping, which can be time-consuming or sensitive to noise (12). To further


63

complicate the problem, any improper combination of the phase data collected with multi-

channel coil will lead to signal cancelation and open-ended fringe lines/cusp artefacts in

the phase images (13-15), which may severely affect the accuracy of phase unwrapping

and, hence, background phase removal. Any error in background phase removal will

propagate into the susceptibility maps (16,17). Consequently, reliable background phase

removal is a critical step, as it directly determines the quality and accuracy of QSM.

Currently, there are mainly four types of background field/phase removal algorithms:

homodyne high-pass filtering (7), geometry dependent artefact correction (GDAC) (8),

projection onto the dipole field (PDF) (9) and sophisticated harmonic artefact reduction for

phase data (SHARP) (10). The homodyne high-pass filtering has been successfully applied

in susceptibility weighted imaging (SWI), due to its robustness and effectiveness.

However, it also causes loss of low spatial frequency phase information to structures with

relatively big size. The high-pass filter size could be reduced in the geometry dependent

artefact correction (GDAC) method, but the accuracy of GDAC is largely dependent on

the a priori information of the geometries of the air-tissue interfaces. Both PDF and

SHARP utilize the fact that the background phase originated from the susceptibility

sources outside the brain satisfies Laplace’s equation (9,10). Particularly, SHARP, or

spherical mean value filtering based techniques have been shown be computationally

efficient in removing the background phase (10). Except for homodyne high-pass filtering,

all of these algorithms require phase unwrapping.

Phase unwrapping could be done through either spatial or temporal domain algorithms

(18,19). For temporal phase unwrapping, multiple echoes are required (19,20). The


64

choices of the echo times may be restricted by the requirement of applying 3D flow

compensation gradient, when studying susceptibilities of veins (7,21). For spatial domain

methods, path-following and quality-guided algorithms are typically used (18,22-24). In

these algorithms, the spatial smoothness of the phase images is assumed. Since the

Laplacian of phase can be calculated directly from the original phase images, another

common strategy is to use Laplacian based phase unwrapping or to use the Laplacian of

phase directly in background field removal (11,17). However, when the Laplacian is

directly used, the accuracy of the extracted local phase information will be reduced (25);

when the Laplacian based phase unwrapping algorithm is used, there are errors in the

regions near the edges of the brain or major veins (11,17,25).

In this study, we propose a background field removal algorithm in which the local phase

unwrapping and background phase removal are performed simultaneously, and the global

phase unwrapping is bypassed. For the best performance of this algorithm, multi-echo

data, in this case double-echo data are used. We also show that the artefacts induced by T2*

effects at a long echo time, as well as the phase singularities can be reduced when handled

properly.

4.2 Theory

For a left-handed system, the phase information in a gradient echo sequence with echo

time TE can be written as:

[4.1],


65

where is the phase offset at TE=0. This phase offset is related to the coil sensitivity

and conductivity of the tissues (26,27). Usually, the original phase images are aliased and

should be unwrapped. This is done by determining and subtracting from the

original phase images:

[4.2].

The unwrapped phase information can be viewed as two components, the background

phase and the local phase :

[4.3].

The background phase and the local phase are directly proportional to the background field

and the local field, respectively. Using the spherical mean value property of the

background field, it was shown that (28):

[4.4],

where “s” is a normalized spherical kernel and * represents the convolution operation.

The background phase can be removed using Eqs. 4.3 and 4.4:

[4.5].

Eq. 4.5 is the critical step to remove the unwanted background phase variations. This is

referred to as spherical mean value filtering (SMV) (28). To calculate the convolution in

Eq. 4.5, there should not be any discontinuities in the original phase images, i.e., phase

unwrapping is required. Denoting the result of Eq. 4.5 as , it is essentially the

difference between the phase value of a given pixel and the mean phase value in the local


66

spherical VOI and, hence, is not dependent on the baseline. Consequently, it is only

necessary to unwrap the local spherical VOI. When the maximal phase difference between

any pixel and the central pixel in the spherical VOI does not exceed π, the local phase

unwrapping can be achieved through baseline shifting:

{ [ ( )]} [4.6].

For each , can be calculated using Eq. 4.5, but only for those pixels which

are sufficiently far away (i.e., with a distance larger than the radius of the spherical kernel)

from any phase wraps. The remnant unresolved pixels are those where the phase gradients

are too big, e.g., the regions close to the air-tissue interfaces, or those which are close to

phase singularities/noisy phase pixels. In the former case, the phase images cannot be

unwrapped locally using baseline shifting and will be set to 0. In the latter case,

for a particular pixel at , was calculated through local complex division:

∑ [4.7],

where N is the total number of pixels of the spherical kernel s centered at .

Combining all the steps, we obtain and then the local phase is extracted via

the deconvolution process:

[4.8].

The regularized deconvolution can be performed in the Fourier domain through

thresholded division or other regularized inversion algorithms (10,29).


67

In the above calculations, it is assumed that TE is sufficiently short or the gradient of total

field variation is sufficiently low. For the phase data at a long TE, TE2, with a

short effective echo time can be constructed through complex division using the phase

images at the shorter TE, TE1:

[4.9],

where is an integer. The choices of is dependent on the TEs and the SNRs of the

phase data. Now can be processed in the same way as described above to obtain

, then can be calculated as:

[4.10].

Finally, the local phase can be solved using through Eq. 4.8.

Alternatively, both and can be generated first, then can be

calculated in a way similar to Eq. 4.10. We refer to the whole process (Eqs. 4.5 to 4.10) as

Local Spherical Mean Value filtering (LSMV) in this thesis.

4.3 Materials and Methods

4.3.1 3D brain model simulation

A 3D brain model was created to study the robustness of LSMV and to compare with

other phase unwrapping algorithms for processing the phase images with single short

echo at different noise levels. Various structures such as basal ganglia structures,

grey/white matter, CSF as well as the air-sinuses were included in this brain model. To


68

simulate the background phase, the susceptibility in the air-sinuses was set to 9.4 ppm.

The phase images were calculated using the forward calculation (8,30–32) at B0=3T,

TE=7.5ms. Magnitude images were also created with different T1, T2*, proton density

values assigned to different structures. White Gaussian noise was added to real and

imaginary parts of the complex data and the SNR of the magnitude in the white matter

region ranged from 5:1 to 20:1. The phase images with different noise levels were

processed using the same steps as described in the in vivo data acquisition and processing

section.

4.3.2 3D cylinder simulation

In order to evaluate the effects of T2* signal decay and partial volume on the proposed

algorithm, the phase images of a 3D cylinder, as surrogate for the vein were created at two

different TEs at B0=3T. The susceptibility inside the cylinder was set to 0.45ppm and

outside 0. The main field direction was set to be perpendicular to the long axis of the

cylinder. The two TEs were chosen to be TE1=7.5ms and TE2=22.5ms. The magnitude

was set to uniformly 1 initially, and then the regions inside the cylinder experienced T2*

signal decay as a function of TE, with T2*=25ms. A background field, varying linearly

along the main field direction, was added to the field map of the cylinder. The gradient of

the background field was chosen such that the induced phase increment is 2π at TE1 within

the full FOV. The partial volume effects were simulated by first creating the magnitude

and phase images of a 2D cylinder in an 8192×8192 matrix, then cropping the central

256×256 pixels of k-space. The radius of the cylinder in the final 256×256 matrix was 8


69

pixels. Finally, the phase images of the 3D cylinder were generated by stacking the phase

images of the 2D cylinder along the long axis of the cylinder. Gaussian noise was added to

real and imaginary channels of the complex data, and the SNR in the magnitude images

varied from 5:1 to 20:1. For phase processing using LSMV, the phase images at TE1 were

complex divided twice into the phase images at TE2 to create phase images with ΔTE of

7.5ms, i.e., in Eq. 4.9. The noisy pixels in the phase images at TE2 were detected as

where phase singularity occurred and where the SNR in magnitude images was less than

2:1. For the deconvolution step in Eq. 4.8, a regularization threshold 0.001 was used. Other

processing steps are the same as described in the in vivo data processing section.

4.3.3 In vivo data acquisition and processing

The proposed method was tested on three in vivo datasets collected with 3D fully flow

compensated double-echo gradient echo sequence (21), with GRAPPA acceleration factor

of 2. Other imaging parameters are listed in Table 4.1. While higher spatial resolution was

used in the first in vivo study, the lower through-plane resolutions used in the second and

third in vivo studies are closer to the situations in most clinical applications. For the

combination of multi-channel data, the linear gradients of the phase images from each

channel were first corrected by shifting the echo center (the position of the pixel which has

the maximum magnitude in k-space) to the actual center of k-space. The baseline of each

channel was estimated from the center of k-space and removed subsequently. Finally, the

phase data were combined through averaging of the complex data weighted by the

magnitude images (13,14).


70

Table 4.1 Imaging parameters for the in vivo data. Datasets 1 and 2 were collected on the

same volunteer.

Dataset 1 Dataset 2 Dataset 3

Echo Time (ms) TE1=7.38, TE2=17.6 TE1=7.38, TE2=22.14 TE1=7.38, TE2=22.14

TR 30 30 30

Flip Angle 15 15 15

Bandwidth (Hz/px) 425 425 425

Head-coil 32-channel 12-channel 32-channel

Voxel size (mm3) 0.6×0.6×1.2 0.5×0.5×2 0.5×0.5×2

Matrix size 512×368×144 512×384×72 512×384×72

The local phase images at TE1 and TE2 were obtained using the following steps.

1. Generating the binary brain masks. This was done using the Brain Extraction Tool

(BET) (33) in FSL, to the magnitude images at TE2. In order to refine the brain masks,

quality maps were generated (24) as a function of the local average number of phase

singularities /poles which can be detected using the method described in (34). The brain

masks were set to 0 where the values of the quality maps were below an empirically

determined threshold.

2. Detecting the noisy pixels in the phase images

There may be noisy pixels inside the brain with unreliable phase values, which could be

induced by the phase singularities or T2* effects at a long TE. To detect these noisy


71

pixels, the phase singularities detected in Step 1 were used as candidates. Then the

noisy pixels were determined as those where the SNR in magnitude image was less than

3:1. These noisy pixels were excluded from the calculation of the SMV filtered results.

3. Spherical mean value filtering (SMV) through global baseline shifting

The SMV filtered results ( ) were first calculated using the original phase images,

only for the pixels which were not affected by any phase wraps in the original phase

images, using Eq. 4.5. The radius of the spherical kernel was set to 8 pixels. The phase

wraps were detected when the absolute value of the phase difference between two

neighboring pixels was greater than 0.9π. The pixels affected by the phase wraps were

determined as those which were within 8 pixels to any phase wraps in 3D. Then the

baseline of all the pixels in the original phase images were shifted by π using Eq. 4.6.

Since most pixels close to the phase wraps in the original phase images have phase

values close to π or – π, a baseline shift of π is most effective in shifting the baseline of

those pixels to 0 and creating locally unwrapped regions. Then SMV filtered results

were obtained using the baseline shifted phase images, using the same procedures as

described above. After this step 3, there might be remnant pixels which are still affected

by phase wraps. These pixels will be resolved in the next step.

4. Spherical mean value (SMV) filtering through local baseline shifting

For a particular pixel, P, the phase value of all the pixels in the spherical VOI centered

at P were complex divided by the phase value of P. After this local baseline shifting, if

no phase wrap was detected in the local spherical VOI, the value of at P will be


72

calculated using Eq. 4.7. If phase wraps were detected, it suggests that either that local

phase gradient is too big, or P is affected phase singularities. In the former case,

cannot be calculated using this proposed method and was set to 0. In the latter case,

was calculated again using Eq. 4.7.

5. Obtaining the local phase information through deconvolution

Finally, the local phase images ( ) were calculated through truncated Fourier domain

division with regularization threshold 0.05, using Eq. 4.8. The whole process is

illustrated in Fig. 4.1.

Figure 4.1 Processing steps in LSMV for single echo phase data with short TE.


73

To generate the local phase images at TE2, the phase images at TE1 were complex divided

twice into those at TE2 to create phase images at . Then was calculated

using steps 2 to 5 as described above, and was calculated as

. For any noisy pixels which were present only in the phase images at but

not in the phase images at TE1, the values of were obtained by projecting the

values of from to as: , assuming that the

influence of the term is sufficiently small.

4.3.4 Comparison with other phase unwrapping algorithms

The proposed algorithm was compared with two commonly used phase unwrapping

algorithms. One is the quality guided 3D phase unwrapping algorithm (3DSRNCP) (23)

and the other the Laplacian based phase unwrapping (18,35). For the comparison, the

phase images were first unwrapped using these two algorithms and then processed using

SHARP. The other parameters, such as the spherical kernel size and the regularization

threshold were the same between SHARP and LSMV. Thus, the differences in the results

were purely caused by the phase unwrapping algorithms. In order to evaluate the

influences of different algorithms on the accuracy of susceptibility quantification,

susceptibility maps were generated using the processed phase images, through truncated k-

space division with a k-space threshold 0.1 (3).

For the simulated 3D brain model and cylinder data, the root-mean-square-errors

(RMSEs) were calculated by comparing the processed phase images with the ideal phase

without background phase or random noise. RMSEs were also calculated for the


74

generated susceptibility maps. For in vivo data, root-mean-square-deviations (RMSDs)

was calculated for both processed phase images and susceptibility maps, by comparing the

results obtained using LSMV and those obtained using the other two phase unwrapping

algorithms. Specifically, the RMSEs for simulated data or RMSDs for in vivo data were

calculated in two regions-of-interest (ROIs): all the regions inside the brain and the regions

close to the veins. For evaluating the processed phase images, the second ROI consisted of

both the regions inside the veins and the surrounding regions within 4 pixels to the veins

(for the 3D cylinder simulation, the second ROI was a cylindrical region with radius 12

pixels, being concentric with the 3D cylinder); while for evaluating the generated

susceptibility maps, the second ROI was taken to be the regions inside the veins (or

cylinder) only. All the tests were performed in MATLAB R2010a on a desktop equipped

with Intel i7-2600 CPU, 16 G of RAM.

4.4 Results

4.4.1 3D brain model simulation results

The RMSEs of the processed phase images generated using different algorithms were

plotted in Fig. 4.2. For both the processed phase images and the susceptibility maps,

LSMV and 3DSRNCP lead to almost the same RMSEs. When SNR was lower than 10,

Laplacian phase unwrapping leads to slightly lower over-all RMSE, but higher RMSE for

the veins, compared with the other two algorithms.


75

Figure 4.2 RMSEs of the processed phase images (a and b) and the susceptibility maps (c

and d) at different noise levels for the simulated 3D brain model. The RMSEs in a and c

were calculated using all the pixels inside the brain, while the RMSEs in b and d were

calculated using only the pixels close to (or inside) the veins. The SNR represents the

signal-to-noise ratio in the magnitude images.

The processing times using different algorithms were also measured, as shown in Fig. 4.3.

For all noise levels, Laplacian phase unwrapping was the fastest among the three tested

algorithms. When SNR was lower than 10, the proposed algorithm LSMV cost more time

than 3DSRNCP. But for higher SNRs, the processing time for LSMV gradually reduced.

This is due to the reduced number of noisy pixels in the phase images. At SNR=20:1, the

processing time of LSMV was roughly 2/3 of that using 3DSRNCP.


76

Figure 4.3 A comparison of the processing times using different algorithms at different

noise levels.

4.4.2 Cylinder simulation results

The RMSEs in the phase images and susceptibility maps of the cylinders processed with

different algorithms are plotted in Fig. 4.4. At both TEs, LSMV led to the smallest RMSEs

among the three algorithms, for both the processed phase images and the susceptibility

maps. When SNR was higher than 8, at both TEs, Laplacian phase unwrapping caused the

largest RMSEs among the three algorithms, for both the processed phase images and the

susceptibility maps. At TE1, 3DSRNCP and LSMV induced similar level of RMSEs. At

TE2, when SNR was lower than 8, phase unwrapping using 3DSRNCP failed and this led

to much larger RMSE in the processed phase images, as shown in Figs. 4.4.b and 4.5. This

error was propagated into the susceptibility maps, as indicated by the large RMSE in the

susceptibility maps at TE2 (Fig. 4.4.d).


77

Figure 4.4 RMSEs in the processed phase images (a and b) and susceptibility maps (c

and d) of the cylinders at two TEs. The SNR represents the signal-to-noise ratio in the

magnitude images.

The original and the processed phase images of the cylinder are shown in Fig. 4.5. When

SNR was 5:1, both 3DSRNCP and Laplacian phase unwrapping failed to unwrap the phase

at TE2. When SNR was 20:1, it was noticed that the 3DSRNCP failed to recover a few

pixels at the edge of the cylinder, as indicated by the black arrows in Fig. 4.5. This is due

to the low SNR at those pixels caused by T2* signal decay. The Laplacian phase

unwrapping caused large error inside the cylinder at TE2 again. On the other hand, for both


78

low and high SNRs, the LSMV algorithm had successfully extracted the local phase

information for the cylinder.

Figure 4.5 The original phase images and the local phase images generated using

different algorithms for the simulated cylinder at SNR=5:1 and SNR=20:1. The SNR

represents the signal-to-noise ratio in the magnitude images. The images in the second to

fourth columns are the central 64×64 pixels in the processed local phase images, as

indicated by the white dashed box in the top-left image. The errors in the local phase

images obtained using 3DSRNCP and Laplacian phase unwrapping are indicated by the

black arrows.


79

4.4.3 In vivo data results

The original and processed phase images, together with the SMV filtered images for

Dataset 1 are shown in Fig. 4.6.

Figure 4.6 The original and processed phase images for Dataset 1. a) Original phase

image at TE1=7.38ms. b) Complex divided phase image with effective TE=2.84ms. c)

Original phase images at TE2=17.6ms. d) SMV filtered result (φSMV) at TE1. e) SMV

filtered result at ΔTE. f) SMV filtered result at TE2, calculated using the images shown in

e and d as e+2×d. g) Local phase image at TE1. h) Local phase image at ΔTE. i) local

phase image at TE2.


80

The LSMV led to almost the same local phase images as those obtained using 3DSRNCP,

as shown in Fig. 4.7. The major difference between LSMV and 3DSRNCP is seen at the

edges of the veins, especially at the longer TE, with the LSMV produced a more

continuous phase profile. This continuous phase profile also helps to reduce the streaking

artefacts in susceptibility maps, as shown in Fig. 4.7.i.


using LSMV (a, d and g) and those generated using 3DSRNCP (b, e and h) for Dataset 1.

The difference images are shown in c, f and i. The phase images in the first row are at

TE1, and the phase images in the second row are at TE2. The images in the third row are

susceptibility maps (SM) obtained using the phase images at TE2 (d and e). The scale bars

are for the difference images c, f and i only. Note the improvement in the SM using

LSMV thanks to the better recovery of phase around the veins.


81


using LSMV (a, d and g), and those generated using Laplacian phase unwrapping (b, e

and h) for Dataset 1. The difference images are shown in c, f and i. The phase images in

the first row are at TE1, and the phase images in the second row are at TE2. The images in

the third row are susceptibility maps obtained using the phase images at TE2 (d and e).

The scale bars are for the difference images c, f and i only.

Compared with the LSMV generated results, the Laplacian phase unwrapping caused

remnant low spatial-frequency background field, especially for regions close to the air-

tissue interfaces, as shown in Fig 4.8. Large differences were also seen at the edges of the

veins. This could be due to the partial volume effects and discretization errors.

The quantitative measures of the differences between different phase processing methods

for all the in vivo data are shown in Table 4.2. For all three datasets, the processed phase

images at TE1 obtained using LSMV are almost the same as those obtained using the


82

3DSRNCP. However, the processed phase images at TE2 show a slight difference between

LSMV and 3DSRNCP. This may be due to the difference of the pixels at the edges of the

veins, caused by T2* signal decay. This difference in the processed phase images was

propagated into the difference in susceptibility maps, especially for the vein regions.

Meanwhile, a relatively large difference was seen between the processed phase images

obtained using Laplacian phase unwrapping and those obtained using LSMV. This also

caused a large difference between the susceptibility maps. As for the processing time, still

Laplacian phase unwrapping with SHARP was the fastest among the three. The time cost

by LSMV for processing the phase images at two TEs was close to the time cost by

3DSRNCP for processing the phase images at a single TE.

The proposed algorithm LSMV is able to handle noisy phase images and reduce the effects

of the cusp artefacts. Signal cancelation and cusp artefacts occur when the multi-channel

phase data is not combined properly. As an example, in Fig. 4.9.a, we show the original

phase images at TE2 from Dataset 2, combined using the built-in algorithm on the scanner.

The improper combination caused cusp artefacts and locally severely reduced SNR on the

phase images, as indicated by the white arrows in Fig. 4.9. The cusp artefact caused failure

of phase unwrapping using 3DSRNCP (Fig. 4.9.b), which was propagated into the

extracted local phase images (Fig. 4.9.e). On the other hand, both LSMV and Laplacian

phase unwrapping handled the cusp artefact properly (Fig. 4.9.d and 4.9.f).


83

Table 4.2 A comparison of phase images and susceptibility maps processed using

different algorithms for the in vivo data.

Dataset 1 Dataset 2 Dataset 3

TE1 TE2 TE1 TE2 TE1 TE2

RMSD in

Phase Images:

Overall (in rad)

LSMV vs.

3DSRNCP+SHARP

0 0.04 0 0.07 0 0.07

LSMV vs.

Laplacian+SHARP

0.05 0.13 0.02 0.09 0.02 0.09

RMSD in Phase

Images: Veins (in

rad)

LSMV vs.

3DSRNCP+SHARP

0.03 0.20 0.04 0.46 0 0.46

LSMV vs.

Laplacian+SHARP

0.06 0.24 0.05 0.47 0.03 0.45

RMSD in

Susceptibility

Maps: Overall (in

ppm)

LSMV vs.

3DSRNCP+SHARP

0 0.01 0 0.01 0 0.01

LSMV vs.

Laplacian+SHARP

0 0.02 0 0.02 0 0.02

RMSD in

Susceptibility

Maps: Veins (in

ppm)

LSMV vs.

3DSRNCP+SHARP

0.02 0.11 0.02 0.18 0 0.15

LSMV vs.

Laplacian+SHARP

0.02 0.12 0.01 0.18 0 0.15

Processing time (s)

LSMV 44 (two TEs) 28(two TEs) 26(two TEs)

3DSRNCP+SHARP 32 31 20 20 20 20

Laplacian+SHARP 7 7 4 4 4 4


84

Figure 4.9 a) Original phase image at TE2 from Dataset 2 with cusp artefact. Note that,

this image was obtained using the built-in multi-channel data combination algorithm on

the scanner. b) Unwrapped phase image using 3DSRNCP. c) Unwrapped phase image

using Laplacian phase unwrapping. d) Local phase image generated using LSMV. e)

Local phase image generated using the unwrapped phase image shown in b. f) Local

phase image obtained using the unwrapped phase image shown in c. Cusp artefact caused

errors in the processed phase images, as indicated by the white arrows.


85

4.5 Discussion

Background field removal in QSM could be affected by several factors. The robustness of

background field removal is usually hampered by the requirement of phase unwrapping,

especially for phase images at a relatively long TE. For short TE phase images, however,

most regions inside the brain are not affected by phase aliasing. Thus the short TE’s phase

images could be used to accelerate the processing of the phase images at a longer TE.

While the shorter TE’s phase has less T2* decay induced noisy pixels, the longer TE’s

phase has higher overall phase SNR. The signal loss due to T2* signal decay could be

reduced by projecting the phase obtained at the shorter TE to a longer TE. It was shown

that the effective magnetic moment is constant (36). Thus, recovering the lost pixels near

the edge of the veins is important, since these pixels will affect the estimation of the

geometries of the veins and consequently affect the estimation of susceptibility. This is the

main advantage of using a combination of short TE and long TE in a double-echo

sequence.

Multi-channel phase data combination also plays an indispensable role in QSM. For phase

data collected with a multi-channel phased array coil, a simple weighted sum of the

complex data usually leads to severe signal cancelation and cusp artefacts in the phase

images (13,14), which will deteriorate the accuracy of background field removal. This

problem is attributed to the coil sensitivity induced phase component in different channel’s

data (13-15), and can be partly solved by applying high-pass filtering to each channel’s

phase data before the data combination. But the combined phase data will be effectively

high-pass filtered and this in turn will lead to under-estimation of the susceptibilities (14).


86

A correction of the constant baseline shifts for different channel’s data is an effective way

to alleviate the problem (13). However, the accuracy of the baseline shift correction is also

dependent on the selection of the reference region that is used for estimating the phase

baseline shifts between channels (14). In a recently proposed method, the coil sensitivity

induced phase components were estimated using reference scan or double-echo data and

subsequently removed before combining the data from different channels (14). Instead of

using the combined phase data, the phase data from individual channels can also be used

directly in QSM processing, followed by weighted average of the susceptibility maps (37).

The latter two methods are more sophisticated, but are computationally more extensive,

since both of them requires phase unwrapping or background field removal for each

channel’s phase data. From our experiences, if the focus is the brain, linear gradients and

baseline correction for each channel’s phase will be sufficient to avoid any cusp artefacts

or signal cancelation in phase images. The selection of a reference channel can be avoided,

if the baseline of each channel is estimated from the center of k-space.

The time cost by background field removal is mainly determined by the phase unwrapping

process. For the currently available phase unwrapping algorithms, the processing time

varies a lot, so does the robustness of these algorithms (12).

The Laplacian phase unwrapping is the most time-efficient. However, no matter whether a

Laplacian operator is directly used to remove the background field, or the phase is

unwrapped first using Laplacian followed by other background field removal algorithms,

errors are usually found in the extracted local phase information, especially for the regions

near the edge of the veins. This is attributed to discretization errors in calculating the


87

Laplacian of phase (11,17,25). Despite its inability in producing reliable phase for the

veins, Laplacian phase unwrapping is usually found to be a robust algorithm, especially

when there are cusp artefacts in the phase images.

On the other hand, the 3D path-following and quality guided phase unwrapping algorithms

are able to produce relatively more reliable phase information, especially for the veins,

even though these phase unwrapping algorithms are usually time-consuming. Particularly,

when there are cusp artefacts, these phase unwrapping algorithms will eventually fail

(15,17), as validated in this study.

The proposed LSMV algorithm leads to almost the same results as those 3D quality guided

phase unwrapping algorithms, with the advantage that phase singularities and noisy pixels

are properly handled. For processing the phase images at two TEs, the time cost by LSMV

is roughly half of the time cost by the fast 3D phase unwrapping algorithm.

The effectiveness of the proposed algorithm is dependent on the TEs used in the double-

echo sequence. A proper combination of the TEs should satisfy the following conditions.

First, the TE of the first echo should be short enough, so that only a few pixels close to the

air-tissue interfaces need to be removed due to too rapid field changes. This helps to

reduce the loss of the regions close to the air-sinuses. Second, the TE of the second echo

should not be too far away from the first TE. i.e., the values of α in Eq. 4.9 should not be

too big. A bigger α means more complex divisions using the phase images at the two TEs,

and hence more amplification of the random noise in the phase images. Under the above

two conditions, and assuming that the effective TE of the complex divided phase,

, we have , or . The lower bound of


88

is usually restricted by the requirement of applying the flow compensation gradient.

This restriction can be removed by using an interleaved double-echo sequence (21), but

with prolonged scan time. In the cases when two relatively long TEs (with short TE

spacing) are used, temporal phase unwrapping can be achieved by projecting the phase at

the short effective TE to a much longer TE, as proposed in former studies (19,20).

However, the accuracy of the processed phase at the longer TE may be compromised by

the amplification of noise due to the projection.

The quality of SHARP processed phase images is also dependent on the deconvolution

process as shown in Eq. 4.8. In this study, we used the truncated k-space division for the

deconvolution, due to its simplicity and time-efficiency. The deconvolution could also be

done through a Tikhonov regularization process (29). It should be noted that the SMV

filtered results generated from this algorithm can be easily used as the input to other

deconvolution algorithms to further improve the accuracy of background field removal.

In addition, the phase component could be used to extract the conductivity of different

tissues (26,27). For the purpose of QSM, the effects of should also be considered, since

it may not satisfy the Laplace’s equation and thus may not be fully suppressed by the

spherical mean value filtering. For the algorithm proposed in this study, the effects of

can be determined through a linear fitting using the processed phase images. Ideally,

should be determined even before the phase images were processed (14). can be

determined through a weighted linear fitting using multiple echoes, but it was also shown

in a previous study that nonlinear fitting of the phase may help to avoid any noise

amplification at longer echo times (16).


89

Finally, the limitations to the proposed method are mainly associated with the regions near

the sinuses, where the field is changing too rapidly. These regions cannot be fully resolved

using baseline shifting or local complex division. Current techniques of using a forward

model to remove more of the air/tissue interface would reduce the rapid field variations

and hence make it possible to apply this new approach even in those regions.

In conclusion, using the proposed algorithm based on double-echo data, global phase

unwrapping is avoided and the noisy pixels in phase images (e.g., phase singularities/cusp

artefacts) are better handled. This improves the efficiency and robustness of background

field removal in QSM.


90

References



NeuroImage 2012;59:2625–35.

2. Li W, Wu B, Batrachenko A, Bancroft-Wu V, Morey RA, Shashi V, et al.

Differential developmental trajectories of magnetic susceptibility in human brain

gray and white matter over the lifespan. Hum. Brain Mapp. 2013;



2010;32:663–76.

4. Fan AP, Bilgic B, Gagnon L, Witzel T, Bhat H, Rosen BR, et al. Quantitative

oxygenation venography from MRI phase. Magn. Reson. Med. 2013; DOI:

10.1002/mrm.24918






Radiology 2012;262:269–78.



8. Neelavalli J, Cheng YN, Jiang J, Haacke EM. Removing background phase

variations in susceptibility‐weighted imaging using a fast, forward‐field calculation.

J. Magn. Reson. Imaging 2009;29:937–48.



NMR Biomed. 2011;24:1129–36.




11. Li W, Avram AV, Wu B, Xiao X, Liu C. Integrated Laplacian-based phase

unwrapping and background phase removal for quantitative susceptibility mapping.

NMR Biomed. 2013; DOI: 10.1002/nbm.3056.

12. Li N, Wang W-T, Sati P, Pham DL, Butman JA. Quantitative assessment of

susceptibility-weighted imaging processing methods. J. Magn. Reson. Imaging 2013; DOI: 10.1002/jmri.24501.




diseases. NeuroImage 2008;39:1682–92.


91



Magn. Reson. Med. 2011;65:1638–48.





susceptibility mapping. Magn. Reson. Med. 2013;69:467–76.



2013;69:1582–94.

18. Ghiglia DC, Pritt MD. Two-dimensional phase unwrapping: theory, algorithms, and

software. Wiley; 1998.

19. Feng W, Neelavalli J, Haacke EM. Catalytic multiecho phase unwrapping scheme

(CAMPUS) in multiecho gradient echo imaging: removing phase wraps on a voxel-

by-voxel basis. Magn. Reson. Med. 2013;70:117–26.

20. Robinson S, Schödl H, Trattnig S. A method for unwrapping highly wrapped multi-

echo phase images at very high field: UMPIRE. Magn. Reson. Med. 2013; doi:

10.1002/mrm.24897

21. Ye Y, Hu J, Wu D, Haacke EM. Noncontrast-enhanced magnetic resonance

angiography and venography imaging with enhanced angiography. J. Magn. Reson.

Imaging 2013;38:1539–48.

22. Jenkinson M. Fast, automated, N‐dimensional phase‐unwrapping algorithm. Magn.

Reson. Med. 2003;49:193–7.



2007;46:6623–35.

24. Witoszynskyj S, Rauscher A, Reichenbach JR, Barth M. Phase unwrapping of MR

images using Phi UN--a fast and robust region growing algorithm. Med. Image Anal.

2009;13:257–68.


spatial variation in tissue composition. NeuroImage 2011;55:1645–56.

26. Haacke EM, Petropoulos LS, Nilges EW, Wu DH. Extraction of conductivity and

permittivity using magnetic resonance imaging. Phys. Med. Biol. 1991;36:723.

27. Kim D-H, Choi N, Gho S-M, Shin J, Liu C. Simultaneous imaging of in vivo

conductivity and susceptibility. Magn. Reson. Med. 2013; doi: 10.1002/mrm.24759

28. Li L, Leigh JS. High-precision mapping of the magnetic field utilizing the harmonic

function mean value property. J. Magn. Reson. 1997 2001;148:442–8.

29. Sun H, Wilman AH. Background field removal using spherical mean value filtering

and Tikhonov regularization. Magn. Reson. Med. 2013; DOI: 10.1002/mrm.24765



Magn. Reson. Part B Magn. Reson. Eng. 2003;19B:26–34.


92




32. Koch KM, Papademetris X, Rothman DL, de Graaf RA. Rapid calculations of

susceptibility-induced magnetostatic field perturbations for in vivo magnetic

resonance. Phys. Med. Biol. 2006;51:6381–402.


55.

34. Cusack R, Papadakis N. New Robust 3-D Phase Unwrapping Algorithms:

Application to Magnetic Field Mapping and Undistorting Echoplanar Images.

NeuroImage 2002;16:754–64.

35. Schofield MA, Zhu Y. Fast phase unwrapping algorithm for interferometric

applications. Opt. Lett. 2003;28:1194–6.

36. Liu S, Neelavalli J, Cheng YCN, Tang J, Mark Haacke E. Quantitative susceptibility

mapping of small objects using volume constraints. Magn. Reson. Med.

2013;69:716–23.

37. Wu B, Li W, Guidon A, Liu C. Whole brain susceptibility mapping using

compressed sensing. Magn. Reson. Med. 2012;67:137–47.


93

Chapter 5 Solving the Inverse Problem

of Susceptibility Quantification

5.1 Susceptibility mapping using truncated k-space division

5.1.1 Theory

As previously discussed in Chapters 2 and 3, the field variation in the main field direction

can be considered as the convolution of the susceptibility distribution and the point-dipole

field response (the Green’s function) as (1,2):

[5.1].

The main field direction is assumed to be in z direction. This convolution can be

calculated through convolution theorem by taking the Fourier transform of Eq. 5.1:

( ) ( ) [5.2].

The Fourier transform of the Green’s function, , has a simple form of ,

when . ( ) , when (1–3).


94

Figure 5.1 The cross-sections of ( ) (a and b) and ( ) (c and d). In a and c, the

cross-sections are parallel to the main field direction, in b and d, the cross-sections are

perpendicular to the main field direction. The white regions in a and b correspond to the

regions where ( ) . The black dashed lines in a and c indicate the positions of

the cross-sections shown in b and d.

Susceptibility quantification is the inverse problem, in which Eq. 5.1 is solved for ,

given . The inverse problem is ill-posed, due to the zeros in the Green’s function

in Fourier domain. This inverse problem can be solved by using a regularized inverse

filter(4), ( ), defined as:


95

( ) { (

)

|

|

(

) (

)

|

|

[5.3].

The cross-sections of ( ) and ( ) are shown in Fig. 5.1.

The regularized inverse filtering leads to two problems: streaking artefacts and

underestimation of the susceptibility (4,5). Consequently, the choice of the regularization

threshold is usually a trade-off between the level of the streaking artefact and the

underestimation of the susceptibility.

The systematic underestimation can be estimated, as proposed in (6). Consider the ideal

solution to the inverse problem as:

( ) ( )

[5.4],

and the solution using the regularized inverse kernel as:

( ) ( )

[5.5],

From Eqs. 5.4 and 5.5,

( )

( ) [5.6].

If the systematic underestimation was considered as a constant scaling factor in image

domain, using the inverse Discrete Fourier Transform, it could be estimated as the

average value of .


96

5.1.2 Optimal choice of the regularization threshold

In order to determine the optimal choice of the regularization threshold (th in Eq. 5.3),

cylinders with different radii ranging from 2 pixels to 16 pixels were simulated in a

256×256 matrix, at B0=3T, TE=5ms. The susceptibility of the cylinders was set to 1 ppm.

Partial volume effects were simulated by first creating the complex data in a 4096×4096

matrix, and then cropping the central 256×256 pixels in k-space. Gaussian noise was

added to real and imaginary channels of the complex data and the SNR in the magnitude

images ranged from 2:1 to 20:1. Means and standard deviations of the susceptibilities of

the both the regions inside and outside the cylinder were measured.

When no random noise was added, the level of underestimation as a function of the

regularization threshold th is plotted in Fig. 5.2.a and 5.2.b, for cylinders with different

radii. When th is small, the predicted underestimation (Figure 5.2.a) agrees better with

the measured underestimation of the relatively bigger cylinders; when th gets bigger, the

predicted underestimation agrees better with the measured underestimation of the smaller

cylinders. The underestimation of the susceptibilities of the cylinders is increasing as the

threshold value th increases, while the uncertainty is decreasing.

When different levels of random noise were added, the levels of underestimation for

different cylinders were almost the same as those when no noise was added, as can be

seen from Figure 5.2.c and 5.2.e. It was also noticed that, for the smallest cylinder, when

th is low, there could be a significant increase of the level of underestimation, as indicated

in Figure 5.2.c. and 5.2.e. The standard deviations of the measured susceptibilities inside


97

the cylinders were increased (Fig. 5.2.d and 5.2.f), compared with the standard deviations

inside the cylinders without any random noise (Fig. 5.2.b).

The measured susceptibilities of the regions outside the cylinders are mainly attributed to

streaking artefacts and Gibbs ringing. As shown in Fig. 5.3.a and 5.3.b, the mean and

standard deviations of the susceptibility values measured outside the cylinders were close

to 0, but the standard deviation can be quite large when random noise was added and

when th is small (Fig. 5.3.dand 5.3.f). The standard deviation measured outside the

cylinders quickly decreases as th increases.

In order to find the optimal threshold value, with the standard deviation outside the

cylinders used as a measure of the level of streaking artefacts, a cost function could be

defined as:

√ [5.7],

where U(th) is the underestimation as a function of th, for cylinder with certain radius r,

and for a certain SNR; N(th) is the standard deviation measured outside the cylinder. The

function R(th) is plotted in Fig. 5.4, for cylinders with different radii. The optimal th can

be selected as the one which minimizes the cost function R(th). The optimal th was found

to be close to 0.1 for small structures and 0.14 for relatively bigger structures. Besides of

underestimation, bigger structures may also lead to blurring of the susceptibility maps

(5,6). Consequently, 0.1 is the over-all optimal th when structures with different sizes are

considered.


98

Figure 5.2 The underestimation in the susceptibility values measured inside the cylinders

with different radii (a, c and e) and the standard deviations (b, d and f). a and b were

generated when no noise was added. For c and d, SNR=10:1 in the magnitude images;

while for e and f, SNR=5:1.


99

Figure 5.3 The mean susceptibility values (a, c and e) and the standard deviations (b, d

and f) measured in the background regions outside the cylinders. a and b were generated

when no noise was added. For c and d, SNR=10:1 in the magnitude images; while for e

and f, SNR=5:1.


100

Figure 5.4 The cost function R(th) for different cylinders. R(th) was calculated using Eq.

5.7. The optimal th was determined as when R(th) was minimized. Particularly, when

SNR=10:1, the optimal ths were 0.12 (r=2px), 0.13(r=4px), 0.14 (r=8px), and 0.14

(r=16px). When SNR=5:1, the optimal ths were 0.1 (r=2px), 0.13(r=4px), 0.13 (r=8px),

and 0.14 (r=16px).

5.2 Geometry constrained iterative reconstruction

5.2.1 Theory and methods

Even though the regularized inverse filtering helps to avoid amplification of the noise, the

streaking artefacts are inevitable and they may severely affect the quality of susceptibility

maps. In order to obtain susceptibility maps with reduced streaking artefacts, additional

information such as the geometries obtained from magnitude or phase images can be used

as constraints, and the inverse problem could be formularized as an optimization process

(7–9):

‖ ( )‖

[5.8],


101

where W is a weighting function and is the regularization function. The first part of

Eq. 5.8 is related to data fidelity and the second part of Eq. 5.8 is related to regularization.

When , Eq. 5.8 reduces to a least-squares fit (3). The is usually taken to be the

Lp-norm the gradients of the susceptibility maps within a low-gradient mask, .

[5.9].

The low-gradient mask could be generated by setting threshold to the gradients found in

magnitude images (7), gradients and Laplacian of the phase images, or a combination of

both the gradients in magnitude and phase images (8). The optimization problem in Eq.

5.8 can be solved using iterative solvers such as LSQR in Matlab (10). Alternatively, a k-

space/image domain iterative algorithm, referred to as iterative SWIM, can be used.

The main idea of iterative SWIM is to reduce the streaking artefacts associated with

structures with relatively high susceptibility values and to update the k-space data in the

singularity regions (i.e., where | ( )| ). The iterative SWIM algorithm consists of

the following steps:

1. Extraction of the high-susceptibility structures from the initial susceptibility maps.

This can be done by setting threshold to the initial susceptibility map, .

A high value of is used at the beginning to extract structures with relatively high

susceptibility values (such as veins in the brain). A denoising filter, e.g., median filter,

is typically applied before extracting the high susceptibility structures through

thresholding. For the extraction of the veins, a high-pass filter could be applied too.


102

This facilitates the suppression of any other bigger structures from being included in

the veins masks. Then the veins can be extracted by thresholding the high-pass filtered

susceptibility maps using . Morphological operations such as opening and closing

were performed after the thresholding to remove the noisy pixels. In the end, a binary

mask is created which equals one for the high susceptibility regions and zero

everywhere else. This mask is applied to the original susceptibility map to get

the extracted susceptibility map . In order to reduce the streaking artefacts

inside the veins, an edge-preserving low-pass filter will be applied, to only the pixels

inside the veins. is Fourier transformed to get .

2. Merging the k-space data. The original k-space data ( ) in the cone of

singularities (i.e. ) is merged with the k-space data . Particularly,

and are merged to get new k-space data as:

( ) { ( ) | ( )|

( ) | ( )|

[5.10],

3. Steps 1 and 2 will be repeated until the relative change of the susceptibility value of

the extracted structures is less than a pre-defined threshold σ. σ was chosen to be 0.01

in this study.

4. Once converged, the threshold will be lowered to include structures with lower

susceptibility values (but still significantly higher than the background), such as the

basal ganglia structures. Then Step 1 to 3 will be repeated.

The whole process is illustrated in Fig. 5.5.


103

Figure 5.5 Illustration of the processing steps of Iterative SWIM algorithm.

5.2.2 Simulations

This algorithm was first evaluated using the 3D brain model. The phase images of this

model were created using the forward calculation at B0=3T and TE=10ms, without any

background field. Uniform magnitude images were used in this simulation. White

Gaussian noise was added to real and imaginary channels of the data and SNR in the final

magnitude images is 20:1. The exact geometries of the veins and other grey matter

structures were used in the iterative algorithm, in order to evaluate the errors purely

caused by the updating of k-space data. First, only the geometries of the veins were input

to the iterative SWIM algorithm. This corresponds to when a high is used to extract

the veins. Next, the geometries of veins and other grey matter structures were input to the

algorithm. This corresponds to when a relatively low is used. The mean and standard

deviations of the susceptibility values of the veins, as well as other grey matter structures


104

(globus pallidus, putamen, caudate nucleus and red nucleus) were measured for each

iteration step. To study the effects of the k-space truncation threshold, different values of

th were tested, with th ranging from 0 to 0.2, with step size 0.01.

5.2.3 In vivo data studies

The iterative SWIM algorithm was also tested using in vivo data collected on a 3T

scanner with TE=14.3ms, voxel size 0.5×0.5×0.5mm3. For this in vivo data, the veins

were extracted with =0.16 ppm. The grey matter structures were extracted using

=0.06 ppm. The accuracy of using these parameters for extracting the geometries was

evaluated using the 3D brain model. Means and standard deviations of the susceptibilities

were measured for veins and grey matter structures including globus pallidus, caudate

nucleus, putamen, red nucleus and substantia nigra. For the veins, the vein masks

extracted in the iterative SWIM algorithm were used for measuring susceptibility, while

for the grey matter structures, manually drawn ROIs were used. The susceptibility maps

generated using the iterative SWIM algorithm were also compared with those generated

using Morphology Enabled Dipole Inversion (MEDI) (7). In MEDI, It is assumed that the

low gradient regions found in magnitude images correspond to the low gradient regions in

susceptibility maps. The nonlinear formulation of MEDI (11) was used, together with

model error reduction through iterative tuning (MERIT). To suppress the gradients

induced by noise in magnitude images, the magnitude images were processed using

anisotropic diffusion filtering (12) with 2 iterations, step size 0.05, gradient threshold 100.

The gradient masks were generated assuming that 30% of the pixels correspond to actual


105

gradients, according to (7). The results of MEDI are largely dependent on the

regularization parameter λ. Note that, the definition of λ in the MEDI algorithm is slightly

different than the “λ” shown in Eq. 5.8. In MEDI, λ is placed together with the data

fidelity term, instead of the regularization term (7,11). Thus, a low λ in MEDI leads to

over-regularized results, and a large λ leads to under-regularized results. In this study, we

tested different values of λ in MEDI, with log10 λ ranging from 1 to 3, with step size 0.25.

The optimal λ was chosen according to the discrepancy principle (13) to match the

residual with the expected noise power, as was used in the previous studies (7,11) on

MEDI.

5.2.4 Simulated Data Results

First of all, the effect of the parameter th is shown in Fig. 5.6, in which the relative errors

in the susceptibility values (measured from the converged susceptibility maps) are plotted

for different structures. As th increases, the relative error in the measured susceptibility

values increases while the standard deviation decreases. In order to reduce the relative

errors and to have low uncertainties in the estimated susceptibility values, in this study we

chose th as 0.1.


106

Figure 5.6 The effects of the parameter th on the accuracies of the estimated

susceptibilities of different structures. a) The relative errors (absolute values) in the

measured mean susceptibilities at different values of th. b) The standard deviations of the

measured susceptibilities at different values of th.

Next, the mean and standard deviations of the susceptibilities of different structures at

different iteration steps were plotted in Fig. 5.7. When only the geometries of the veins

were used as constraints, the relative error in the mean susceptibility value of the veins

quickly decreases and converges after the 3rd

iteration, while the relative errors for all the

other structures were increasing, as shown in Fig. 5.7.a. When the geometries of all the

structures, including the veins, were used in the iterative SWIM algorithm, reduced

relative errors in the measured susceptibility values were observed for all the structures,

as indicated in Fig. 5.7.b. For the veins, the relative error was maintained at the same

level as in Fig. 5.7.a. The standard deviations were reducing as more iteration steps were

conducted, as can be seen from Figure 5.7.c to 5.7.f.


107

Figure 5.7 Means and standard deviations of the susceptibility values of different

structures in the 3D brain model at different iteration steps. a and b: the mean

susceptibility values measured at different iteration steps. c to f: the standard deviations

measured at different iteration steps. The images in the first column (a, c and e) show the

changes in mean and standard deviation when a high χth was used in geometry extraction,

while the images in the second column show the results when a low χth was used in

geometry extraction.


108

The iterative SWIM algorithm has reduced the streaking artefacts effectively, as can be

seen from Fig. 5.8. Nonetheless, this iterative algorithm is optimized for the veins and the

selected grey matter structures only. For the other parts of the brain, the relative errors in

the measured mean susceptibility values were slightly increased after the iterative SWIM

algorithm, as indicated by the error maps shown in Fig. 5.8.c and 5.8.d.

Figure 5.8 a) The initial susceptibility map. b) The final susceptibility map after iterative

SWIM. c) The difference between a and the ideal susceptibility map. d) The difference

between b and the ideal susceptibility map.


109

Figure 5.9 K-space profiles of the initial susceptibility map (a) and the final

susceptibility map after iterative SWIM algorithm (b). The errors in the k-space profiles

in a and b are shown in c and d, respectively. Specifically, the errors were calculated by

comparing the k-space of the generated susceptibility maps with the kspace of the ideal

susceptibility maps. Compared with the initial susceptibility maps, the final susceptibility

maps after iterative SWIM algorithm have reduced errors in the cone of singularities in k-

space, as indicated by the white arrows.

The k-space profiles of the original susceptibility map and the final susceptibility map

after iterative SWIM algorithm are shown in Fig. 5.9. The errors of k-space data in the

cone of singularity regions were reduced after the iterative SWIM algorithm, especially

for the central part of the k-space, as indicated by the white arrows in Fig. 5.9.


110

5.2.5 In vivo Data Results

Figure 5.10 The extracted geometries of veins and grey matter structures (a and b) and

the susceptibility maps (c and d) for the in vivo data. a) Maximum intensity projection of

the binary masks of veins extracted using a high χth. b) One slice of the binary masks of

the grey matter structures extracted using a low χth. c) Maximum intensity projection of

the initial susceptibility maps. d) The corresponding susceptibility map to the binary

mask shown in b.

The vein masks obtained from the in vivo data are shown in Fig. 5.10. Particularly, Fig.

5.10.a shows the maximum intensity projection (MIP) of the binary vein masks obtained

from single high-pass filtering and thresholding, while Fig. 5.10.b shows the one slice of

the binary masks for other grey matter structures with lower susceptibility values than the


111

veins. In order to evaluate the accuracy in the structure extraction, the vein masks and

grey matter structure masks were generated for the brain model using the same algorithms

and parameters. The relative error (calculated as (number of missed pixels +number of

false positives) / actual number pixels in veins) in the extracted veins was measured as 9%

and 10%, respectively. For the in vivo data, it was also observed that, the detected

geometries of the veins or grey matter structures were sometimes locally contaminated by

the noise and streaking artefacts which may have high intensities on the susceptibility

maps. Besides, the putamen was only partly extracted, as shown in Fig. 5.10.b. This

could be due to the variation of susceptibilities within the putamen.

Similar to what was seen in the simulation, in the first round of iterations, when the

geometries of the veins were extracted using the high , the mean susceptibility values

of veins quickly converge after the 3rd

iteration step, as shown in Fig. 5.11.a. The

susceptibility values of other structures were decreased after the iterative algorithm was

applied, except for red nucleus, for which the susceptibility value was slightly increased,

as indicated by the images in the first columns of Figs. 5.11 and 5.12. In the second round

of iterations, when the geometries of both the grey matter structures and the veins were

extracted using a low , the mean susceptibility values of all the grey matter structures

were increased, while the susceptibility of the veins was maintained at the same level as

in the first round of iterations, as can be seen from the second columns of Figs. 5.11 and

5.12. For the veins, the standard deviations in the measured susceptibilities were

decreased as more iteration steps were conducted, as shown in Fig. 5.13.a and 5.13.b. For

other structures, the standard deviations measured in the second round of iterations were


112

slightly increased (Fig. 5.13.c and 5.13.d). This could be caused by the errors in the

extracted geometries of these structures. The streaking artefacts were reduced

dramatically for all the structures, as can be seen from Fig. 5.14.

Figure 5.11 Mean susceptibility values of veins and globus pallidus at different iteration

steps in the in vivo data. The figures in the first column were obtained when high χth was

used to extract the geometries of veins only, while the figures in the second column were

measured when low χth was used to extract the geometries of veins and other structures.


113

Figure 5.12 Mean susceptibility values of red nucleus (RN), caudate nucleus (CN) and

putamen (PUT) at different iteration steps in the in vivo data. The figures in the first

column were obtained when high χth was used to extract the geometries of veins only,

while the figures in the second column were measured when low χth was used to extract

the geometries of veins and other structures.


114

Figure 5.13 Standard deviations of the susceptibility values at different iteration steps for

different structures in the in vivo data. The figures in the first column were obtained when

high χth was used to extract the geometries of veins only, while the figures in the second

column were measured when low χth was used to extract the geometries of veins and other

structures.


115

Figure 5.14 Comparison between the initial susceptibiltiy maps (a and d) and the final

susceptibility maps (b and e). Their differences are shown in c and f (generated as b-a

and e-d).

5.2.6 Comparison with Morphology Enabled Dipole Inversion (MEDI)

For MEDI, the proper value of λ was selected based on Fig. 5.15. The susceptibility maps

generated with different values of λ are shown in Fig. 5.16. The smaller the λ, the more

smoothed the susceptibility maps, and the smaller the RMSE. In this study, the optimal λ

was chosen to be 100.

Compared with the susceptibility maps generated by iterative SWIM algorithm, MEDI

leads to smoother images with preserved edge information in the predefined regions.

However, since MEDI utilizes the edge information in magnitude images only, when

there is no clear edge, MEDI may lead to underestimation of the susceptibilities of veins,

as shown in Fig. 5.17. In the cortical region, these two methods yield similar results.


116

Figure 5.15 The effects of the regularization parameter λ in MEDI. The average gradient

was measured in the low gradient regions in the converged susceptibility maps.

Figure 5.16 Susceptibility maps generated using different values of regularization

parameter λ with MEDI. a and d: λ=10. b and e: λ=100. a and d: λ=1000.


117

Figure 5.17 Comparison of susceptibility maps generated by iterative SWIM (a) and

MEDI (b). The profiles of the solid black lines across the vein of Galen in a and b are

shown in c. The profiles of the dashed black line in the cortical region were shown in d.

The corresponding magnitude image is shown in e, which does not have clear edge

information of the veins in the region indicated by the white arrow.


118

This is further confirmed by the histograms of the susceptibility values of the veins shown

in Fig. 5.18. Compared with MEDI, iterative SWIM algorithm leads to higher

susceptibility estimates of the veins. Meanwhile, the iterative SWIM provides almost the

same susceptibility estimates for grey matter structures such as globus pallidus, as shown

in Table 5.1. The distribution of the susceptibility values of the pixels inside the globus

pallidus is narrower in MEDI than in iterative SWIM. This is due to the minimization of

the edges inside globus pallidus in the susceptibility maps generated by MEDI.

Figure 5.18 Distributions of the susceptibility values of the pixels inside the veins (a and

b) and inside the globus pallidus (c and d), measured from iterative SWIM generated

susceptibility maps (a and c) and from MEDI generated susceptibility maps (b and d).

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 10

2000

4000

6000

8000iterative SWIM: veins

Susceptibility /ppm

Nu

mb

er

of p

ixe

ls

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 10

2000

4000

6000

8000MEDI: veins

Susceptibility /ppm

Nu

mb

er

of p

ixe

ls

-0.2 -0.1 0 0.1 0.2 0.3 0.40

2000

4000

6000

8000

iterative SWIM: GP

Susceptibility /ppm

Nu

mb

er

of p

ixe

ls

-0.2 -0.1 0 0.1 0.2 0.3 0.40

2000

4000

6000

8000

MEDI: GP

Susceptibility /ppm

Nu

mb

er

of p

ixe

ls

a b

c d


119

Table 5.1 Mean and standard deviation (in ppm) measured from susceptibility maps

generated using iterative SWIM algorithm and MEDI.

Iterative SWIM MEDI

Vein 0.30±0.11 0.24±0.15

Globus Pallidus 0.11±0.06 0.12±0.04

Red Nucleus 0.05±0.05 0.06±0.03

Caudate Nucleus 0.04±0.05 0.04±0.03

Putamen 0.02±0.05 0.03±0.04

The k-space profiles of the susceptibility maps are shown in Fig. 5.19. The major

difference between the k-space obtained from MEDI and the k-space generated by

iterative SWIM is associated with the singularity regions in k-space. For the k-space

generated using iterative SWIM algorithm, the lower amplitude inside the singularity

regions is due to the fact that only veins and certain grey matter structures with high

susceptibility values were used to update the k-space. It can also be observed that outside

the singularity regions, the amplitude of the k-space generated by iterative SWIM is

slightly higher than that generated by MEDI, as indicated by the white arrow in Fig.

5.19.d. This is due to the difference between MEDI and iterative SWIM in defining the

cone of singularities. In iterative SWIM, the cone of singularities is explicitly defined

using the k-space truncation threshold, and only the k-space data inside this singularity

region are updated.


120

Figure 5.19 The k-space profiles obtained using truncated k-space division (a), iterative

SWIM (b) and MEDI (c). The difference between b and c is shown in d. The window

levels in a, b, and c are the same. The major differences are seen along the cone of

singularities in k-space, as indicated by the white arrow.

5.3 Noise in susceptibility mapping

The noise in susceptibility mapping is dependent on several factors, such as the

regularization in solving the inverse problem and the random noise in the original phase

data. In order to study the propagation of the random noise in the original phase data to

the noise in susceptibility maps, simulations were conducted using 2D cylinders with

different radii ranging from 2 to 16 pixels. The phase images were created at B=3T,

TE=5ms. The susceptibility value inside the cylinder was set to 1, outside 0. A constant

value was assumed to create magnitude images. To simulate the partial volume effects,

the magnitude and phase images were first created in a 4096×4096 matrix, then the

central 256×256 of k-space were cropped to form the final magnitude and phase images.


121

Random noise was added to real and imaginary channels of the complex data, and the

SNR in the magnitude images ranged from 2:1 to 20:1.

The susceptibility maps were generated using truncated k-space division, with k-space

truncation threshold th ranging from 0.01 to 0.66, with step size 0.01. In order to study

the propagation of random noise, independent of different regularization algorithms, the

singularity regions where were replaced using the k-space data of the ideal

susceptibility maps. Means and standard deviations of susceptibility values were

measured from both the regions inside the cylinders and a referenced region in the

background. The mean susceptibility values were used to calculate the relative error in

susceptibility estimation using different regularization thresholds with different levels of

noise, while the standard deviations were used to estimate the noise in the susceptibility

maps.

As shown in Fig. 5.20.a and b, the standard deviations measured in the reference region

in the background are consistent for different cylinders. However, the standard deviations

measured inside the cylinders show large variation. For large cylinders, the standard

deviations measured in the two different regions are similar. Thus, the large difference

between the standard deviations in Fig. 5.20.a and b for small cylinders is mainly due to

the partial volume effects. When the standard deviations measured in the background

reference region was used in analysis, the noise in susceptibility maps is found to be

dependent on the noise in the original phase images. This relation can be approximated as

a linear function as:


122

[5.11],

where represents the noise amplification factor. As shown in Fig. 5.20.c, when the

regularization threshold th was chosen to be 0.1, is approximately 4. This is consistent

with the amplification factor found in the previous study (14). Note that gets smaller as

the regularization threshold th increases, although the relative error will also increase as

th increases, as discussed in earlier sections. When was plotted as a function of the

relative error in the estimated susceptibilities, there appears to be an optimal

regularization threshold th=0.13 which leads to both low noise amplification and low

relative error in the susceptibility estimation. This is consistent with the optimal

regularization threshold found earlier in this chapter.


123

Figure 5.20 a) Standard deviation of susceptibilities measured in the background

reference region for cylinders with different radii. b) Standard deviation measured inside

the cylinders. c) Noise amplification factor α as a function of SNR in magnitude images,

when th=0.1. d) Noise amplification factor α as a function of relative errors in the

estimated susceptibilities, when SNR=10:1. The arrow shows the case when th=0.13,

α=3.3 and the underestimation of the susceptibility is 11.3%.


124

5.4 Discussion and Conclusions

The main obstacle of quantitative susceptibility mapping originates from the ill-posed

inverse problem. This leads to mainly three problems: underestimation of the

susceptibility, streaking artefacts, and amplification of the noise.

The underestimation of susceptibility can be well-approximated for structures with

relatively large susceptibility values, such as veins. But for other structures, such as the

grey matter structures, the estimation of susceptibilities is affected by the streaking

artefacts caused by the nearby strong susceptibility structures. In this case, using a scaling

factor to reduce the underestimation may lead to large uncertainties. Furthermore, it was

noticed that when a large k-space truncation threshold was used, there tends to be more

blurring of the susceptibility maps (4,5).

Another problem of current susceptibility mapping technique is related to the streaking

artefacts. The inverse problem can be efficiently solved in Fourier domain through

truncated k-space division, and the streaking artefacts are related to the regularization that

is applied in the k-space division. Newer techniques aim to map the susceptibilities as an

optimization problem, and impose different constraints to the inverse problem. The

accuracies of these newer methods are largely dependent on the constraints that are used

in the optimization. It has already been shown that purely the gradient information

extracted from magnitude images may not be sufficient, as not all the edges in

susceptibility maps correspond to detectable edges in the magnitude images (8). This is

partly solved by integrating the gradient information found in both magnitude and phase


125

images (8). Essentially, all of these methods fall into the category of edge preserving

smoothing. Even though the susceptibility maps generated by these algorithms are smooth,

the computation time can be prohibitively long. The k-space/image domain iterative

SWIM algorithm, on the other hand, is much faster and more suitable for clinical settings.

For the in vivo data in this study, which has 512x512x256 matrix size, it took MEDI 1.9

hours to converge, but merely 1.6 minutes for iterative SWIM algorithm, on the same

desktop with Intel i7 CPU.

The effectiveness of the iterative SWIM algorithm was also examined using simulated

data and in vivo data. The iterative SWIM algorithm has replaced the original k-space

data in the cone of singularities, using the k-space of high-susceptibility structures. This

results in proper interpolation of the missing data in central part of the k-space, and

suppression of the noise with high frequency due to the replacing of missing data in the

periphery of k-space. The iterative SWIM algorithm preserves the susceptibility

information of the veins, but may potentially under-estimate the susceptibility of bigger

grey matter structures, if they are not included in the geometry masks. Nonetheless, the

streaking artefacts surrounding the veins were largely reduced together with improved

accuracy of estimating the susceptibility of the veins. This is particularly advantageous

for studies related to measuring venous oxygen saturation.

Finally, the propagation of the random noise from original phase images to susceptibility

maps was studied. With typical truncation threshold of 0.1, the noise amplification factor

is close to four. This amplification factor was obtained assuming that only the cone of

singularities in k-space was updated or modified. There are also studies on the noise


126

behavior in susceptibility mapping, using algorithms that update all the k-space values

(15). Due to the smoothing function used in these algorithms, it is expected that the noise

will be reduced in susceptibility maps.

The accuracy of susceptibility mapping is also affected by many other factors,

particularly, partial volume effects. While this is less a problem for relatively big

structures such as the globus pallidus, with the resolution used clinically (e.g., 1mm x

1mm x 2mm), it may cause errors for the veins. Considering a vein with diameter close to

the imaging resolution, it may be still feasible to obtain a susceptibility value for the pixel

which contains the vein. However, due to partial volume effects, it is not possible to

obtain an accurate susceptibility value for the vein itself, but only an estimate of the

magnetic moment, unless certain a priori information about the size of the vein is

available (16,17).

In conclusion, the core of quantitative susceptibility mapping is solving the inverse

problem. Various algorithms have been proposed to deal with the ill-posedness of this

inverse problem. The k-space/image domain iterative algorithm is particularly time-

efficient and applicable, especially for studies focused on measuring venous oxygen

saturation.


127

References



Magn. Reson. Part B Magn. Reson. Eng. 2003;19B:26–34.






2009;54:1169–89.



2010;32:663–76.



Med. 2009;62:1510–22.



2013;69:1582–94.




2012;59:2560–8.



NeuroImage 2012;62:2083–100.



NeuroImage 2012;59:2625–35.

10. Paige CC, Saunders MA. LSQR: An Algorithm for Sparse Linear Equations and

Sparse Least Squares. ACM Trans Math Softw 1982;8:43–71.



susceptibility mapping. Magn. Reson. Med. 2013;69:467–76.

12. Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE

Trans. Pattern Anal. Mach. Intell. 1990;12:629–39.

13. Morozov, VA. On the solution of functional equations by the method of

regularization. Sov. Math Dokl 1966;7:414–7.





128

15. Liu T, Xu W, Spincemaille P, Avestimehr AS, Wang Y. Accuracy of the

morphology enabled dipole inversion (MEDI) algorithm for quantitative

susceptibility mapping in MRI. IEEE Trans. Med. Imaging 2012;31:816–24.

16. Cheng Y-C, Hsieh C-Y, Neelavalli J, Haacke EM. Quantifying effective magnetic

moments of narrow cylindrical objects in MRI. Phys. Med. Biol. 2009;54:7025–44.



Med. 2013;69:716–23.


129

Chapter 6 Improved Venography

using True Susceptibility Weighted

Imaging (tSWI)2

6.1 Introduction

Susceptibility weighted imaging (SWI) is a high resolution, spoiled gradient echo (GRE)

magnetic resonance imaging (MRI) technique used today clinically for evaluating small

veins and venous abnormalities in the brain and the presence of increased iron content

and microbleeds in diseases such as dementia, multiple sclerosis (MS), Parkinson’s

disease, stroke and traumatic brain injury (TBI) (1–3). SWI's exquisite sensitivity to small

tissue magnetic susceptibility changes is due to its use of phase information (3–5).

Paramagnetic or diamagnetic substances relative to water such as blood products or

calcium, respectively, perturb the local magnetic field proportional to their respective

magnetic susceptibilities. These differences are reflected in the phase of the MR images.

2Most of the contents in this chapter are adapted from Liu S, Mok K, Neelavalli J, Cheng YCN, Tang J, Ye

Y, Haacke EM. Improved MR Venography Using Quantitative Susceptibility-Weighted Imaging. J. Magn.

Reson. Imaging 2013; Article first published online: 31 OCT 2013 DOI: 10.1002/jmri.24413. Reprint under

license.


130

In conventional SWI, after applying appropriate unwrapping and/or filtering techniques

on the raw phase data (1,3), a phase dependent mask is created and multiplied n times into

the magnitude data to enhance the contrast/visibility of these substances.

Although SWI has been used quite successfully in clinical applications for many years, it

is important to realize that it has a few weaknesses. One of them is based on the fact that

the MRI phase signal is not only a function of the susceptibility, but also dependent on

shape and orientation of the structure of interest. In data acquired with sufficient

resolution, the phase inside veins perpendicular to the field has the opposite sign to that

inside veins parallel to B0. This leads to variable suppression effects with the phase mask

that makes SWI unique over conventional gradient echo imaging (6). Recently,

quantitative susceptibility mapping (QSM) has emerged as a means to extract the source

of phase information, that is, the local susceptibility distribution (7–15). QSM is known to

be independent of echo time and, to a large degree, of orientation. To avoid the vessel

orientation dependence in routine SWI data, instead of phase, we propose using a mask

based on the susceptibility map. We refer to this approach as true-SWI (tSWI) to

distinguish it from the conventional phase mask based SWI. In this work, our purpose is

to compare the ability of these two methods to improve venous contrast and to show that

tSWI is able to remove the geometry dependence of the phase for veins and microbleeds

in SWI data.


131

6.2 Materials and Methods

To provide some flexibility in generating the susceptibility weighting mask W, we

introduce both lower and upper thresholds for defining the mask as follows:

{

[6.1],

where refers to the susceptibility value of a voxel (e.g., vein) relative to the surrounding

tissue in the susceptibility map, is the lower limit and is the upper limit of the range

of tissue susceptibility values for which we want to improve the contrast in the final

susceptibility weighted image. Finally, the true-SWI is generated by multiplying the

magnitude image with the mask n times similar to the usual SWI mask application:

[6.2].

The whole process is illustrated in Fig.6.1.

In order to determine appropriate values for these two thresholds, we examined different

potential choices. For , we used: 1) the mean susceptibility value in the background

white matter tissue region (0 ppm) in the susceptibility map and 2) three standard

deviations ( ) above the tissue region, where is the standard deviation of the white

matter tissue region in the susceptibility map. While a threshold of 0 ppm would ensure

that the susceptibility weighting mask would include smaller veins that are partial

volumed, it can also lead to increased noise in tissue regions where susceptibility is

supposed to be zero. On the other hand, a choice of would reduce inclusion of

noise in the mask. For , we used: 1) the expected mean susceptibility value in the vein,


132

which is about 0.45 parts per million (ppm) relative to water (3,16) under normal

physiological conditions; and 2) a value higher than this, in this case 1ppm.

Figure 6.1 A comparison between tSWI and SWI data processing steps.


133

For a given set of and values, the contrast-to-noise ratio (CNR) between the vein

and the background tissue can be optimized by choosing an appropriate n value in Eq. 6.2.

CNR for a vein can be defined as the ratio between tSWI signal contrast for the vein and

its associated uncertainty as follows:

[6.3],

where √

. Noise in either the reference (background tissue)

region ( ) or in the vein ( ) in the tSWI image can be estimated using the

following equation:

√

[6.4].

The noise, represented by the standard deviation for a given voxel in W, , is dependent

on the noise, , in the susceptibility map in the following manner:

{

[6.5].

The factor 1/2 in the case when or in Eq. 6.5 is due to the discontinuity (1).

For simplicity, we assume that signal from vein (or cylinder) and reference region in the

original magnitude images are the same: , and their associated

signal standard deviations in these two regions are also the same:

. Furthermore, we assume that the mean susceptibility value of the reference region is 0,

and the mean susceptibility value of the vein is .


134

i) When a threshold of = 0 was used to generate the susceptibility mask, in the

reference region, W = 1, and . In the vein, W = , ,

for ; but when , W = 0, ; when , W =

0, = 0. Using Eq. 6.4,

{

(

(

)

(

)

)

(

)

(

)

[6.6].

ii) When a threshold of was chosen, in the reference region, W = 1, and = 0

due to the fact that most pixels in the reference (background tissue) region have

susceptibility values less than . In the vein, W = ,

, for ; but when , W = 0, ;

when , W = 0, = 0. Thus,

{

( (

)

(

)

( ) )

(

( ) )

[6.7].

Thus the CNR can be written as:


135

(

)

[6.8],

for , with given in Eqs. 6.6 and 6.7.

When and is slightly less than , CNR approaches SNR in the magnitude

images as n approaches infinity. In this case, the optimal n was chosen to be the value

where CNR reaches 90% of the maximal CNR for a certain vein.

Simulations

To evaluate the theoretical predictions, the optimal choice of n for generating tSWI

images for different threshold values and vessel susceptibility values, and the influence of

high-pass filtering on the final CNR for veins in tSWI images, simulations were

performed using cylinders as surrogates to veins. A series of cylinders with radii ranging

from 2 pixels to 16 pixels was used to simulate the associated phase images in a 512×512

matrix at B0=3T, and TE=10ms. The cylinders were taken to be perpendicular to the main

magnetic field. The input susceptibility of the cylinders was set to be 0.45ppm and the

susceptibility value of the background region was set to zero. In order to simulate a more

realistic response of the field perturbation, the complex data of a cylinder with radius 16

times of the final radius was first created on an 8192×8192 matrix. The magnitude signal

for the cylinder and background region were taken to be unity. The central 512×512 k-

space points generated from the larger matrix were then used to reconstruct the complex

images of the cylinders. Gaussian noise was added to both real and imaginary channels of

the data to simulate the SNR in the magnitude images to be 10:1. The simulated phase

images were processed using a homodyne high-pass filter with a k-space window size of


136

64×64. Two sets of susceptibility maps, one from unfiltered and the other from filtered

phase images, were generated for each cylinder size, using truncated k-space division

with a k-space threshold of 0.1 (7). This is to evaluate the influence of high-pass filtering

on the final CNR in tSWI images. The tSWI images were generated using Eqs. 6.1 and

6.2 for different values of and as mentioned in the previous section. The standard

deviation of the susceptibility maps was measured from a reference region outside the

cylinder. The susceptibility mask was multiplied into the magnitude image n times with n

ranging from 0 to 10 (n=0 refers to the case of no mask multiplication). The local CNRs

between cylinders and the background reference were measured from the tSWI data using:

[6.9]

where Svein and Sref are the mean intensity values inside the cylinder (vein) and inside a

reference region of interest (ROI) adjacent to the cylinder directly from tSWI image,

respectively. In order to estimate the overall noise directly from tSWI images, the

standard deviations inside the cylinder ( ) and the reference region ( ) in

tSWI were measured and was again calculated as the square root of

. The theoretically predicted CNRs from Eq. 6.8 using different thresholds for

generating the susceptibility mask were compared with those measured from the

simulations and the appropriate value for for processing in vivo data was determined.

CNRs of the cylinders with different susceptibility values ranging from 0.2ppm to

0.45ppm were calculated to evaluate the influence of the susceptibility value of the object

on the optimal choice of n.


137

In vivo data

Table 6.1 Imaging parameters for three volunteers and one patient for in vivo studies.

Dataset 5 was collected on a TBI patient.

To evaluate the efficacy of tSWI in in vivo neuro-imaging, we compared the CNR

obtained in tSWI data with that obtained in conventional SWI images in three healthy

adult volunteers. The study was approved by the local institutional review board and

informed consent was obtained from all subjects before the MRI scan. The volunteers

were imaged on a 3T Verio system (Siemens, Erlangen) using a 3D SWI sequence with

isotropic voxel size of 0.5mm×0.5mm×0.5mm. Imaging parameters are given in Table

6.1. Data were acquired in the transverse orientation. In one case (volunteer 1), the SWI

sequence was performed twice using two different echo times (TE = 14.3ms and 17.3ms).

To evaluate the influence of voxel aspect ratio on the CNR, lower resolution images of

0.5mm×0.5mm×2mm (anisotropic voxel size) from all 4 volunteer datasets were

Dataset No. 1 2 3 4 5

Volunteer No. 1 1 2 3 -

B0 (T) 3 3 3 3 3

TR (ms) 26 26 24 24 29

TE (ms) 14.3 17.3 17 15.3 20

FA (degrees) 15 15 15 12 15

BW (Hz/px) 121 121 181 121 120

Voxel size (mm3)

0.5×0.5

×0.5

0.5×0.5

×0.5

0.5×0.5

×0.5

0.5×0.5

×0.5

0.5×0.5

×2

Matrix Size 512×368

×256

512×368

×256

512×368

×224

512×368

×192

512×416

×64


138

generated by taking the central portion of the original k-space along the transverse

direction.

The quantitative susceptibility maps were generated for the isotropic and the anisotropic

data, as follows: tissues outside the brain were removed using the Brain Extraction Tool

(BET) in FSL (17), a homodyne high-pass filter with a k-space window of 64×64 was

applied to remove the background field induced phase artefacts (1), and the inversion

process to create susceptibility maps was accomplished using a single orientation

truncated k-space division approach with a k-space truncation threshold of 0.1 (7).

Similar to the case of the simulated data, two sets of tSWI images for each of the SWI

datasets were generated using Eqs. 6.1 and 6.2 with: a) =0 and b) , where

is the standard deviation of the susceptibility in the background white matter region close

to the vein for which the CNR was measured. The threshold was kept at 0.45ppm. The

susceptibility mask was multiplied into the magnitude image n times, with n ranging from

1 to 10. To generate conventional SWI images, phase masks were created using the high-

pass filtered phase images and then multiplied four times into the magnitude images (1).

To investigate the impact of newer data processing methods, we also applied phase

unwrapping (18) and SHARP (9) to remove the background field, and applied a geometry

constrained iterative algorithm (8) to reconstruct the susceptibility maps for one dataset

(Dataset 2). Then, tSWI images were generated using =0 and =0.45ppm. Local

CNRs of two selected veins, the left internal cerebral vein (LICV) and the right septal

vein (RSV), were measured from both tSWI and SWI using Eq. 6.9. Each vessel’s ROI

was selected from the susceptibility maps and copied onto the tSWI or SWI images for


139

CNR evaluation. The reference ROI adjacent to each vein was taken from the same slice.

To demonstrate the advantages of using tSWI over conventional SWI, we also analyzed

one SWI dataset from a TBI patient. For this patient dataset, susceptibility maps were

generated through homodyne high-pass filtering and truncated k-space division. tSWI

images were then obtained with susceptibility weighting masks generated using =0,

=0.45ppm and n=2. All the processing was done using Matlab (R2010a, Natick, MA).

6.3 Results

Simulations

The simulated phase images of cylinders of different sizes, their corresponding

susceptibility maps and tSWI images are shown in Fig. 6.2. The measured CNRs for

cylinders with different radii, but with a constant input susceptibility of 0.45ppm, are

plotted as a function of n in Fig. 6.3, and the theoretically predicted CNRs are plotted in

Fig. 6.4. Since no T2* effects are considered here, the CNRs shown in Figs. 6.3 and 6.4

reflect contrasts from only phase/susceptibility differences between the cylinders and the

background reference region. The optimal choice of n and, correspondingly, the value of

CNR in the tSWI image, are influenced both by (a) the choices of and and (b) the

high pass filter. For =0, CNRs reach maximum when n ≤ 4 (Figs. 6.3.a, 6.3.c, 6.3.e,

6.4.a and 6.4.c). When is larger with =0, it also takes a larger n value to reach the

optimized CNR. Meanwhile, the choices of and can also affect the rate at which

optimal CNR is approached as a function of n. For , CNRs in general increase

as n increases (Figs. 6.3.b, 6.3.d, 6.3.f, 6.4.b and 6.4.d). The optimal n may be chosen


140

when CNR reaches 90% of the maximum of CNR. The optimal n was 4 for =0.45ppm,

and greater than 10 for =1ppm.

Figure 6.2 Phase images (a and b), susceptibility maps (c and d) and tSWI images (e, f, g

and h) for simulated cylinders with and without homodyne high-pass filtering. Images in

the first and third columns are generated using the original phase images without high-

pass filtering, while images in the second and fourth columns are generated using high-

pass filtering. The tSWI images e and f were generated using χ1=0, χ2=0.45ppm, n=2;

while g and h were generated using χ1=3σχ, χ2=0.45ppm, n=4. σχ is the standard deviation

of a reference region measured from the susceptibility maps shown in c and d

(σχ=0.05ppm for both c and d). The SNR in the original magnitude image was set to be

10:1 and the CNR between the cylinders and background in the original magnitude

images was basically zero.


141

Figure 6.3 Measured CNRs of cylinders from simulated tSWI images. Figures in

different rows were generated using different χ2 values, while figures in different columns

were generated using different χ1 values. a) χ1=0, χ2=1ppm; b) χ1=3σχ, χ2=1ppm; c) χ1=0,

χ2=0.45ppm; d) χ1=3σχ, χ2=0.45ppm; e) χ1=0, χ2=0.45ppm; and f) χ1=3σχ, χ2=0.45ppm.

To evaluate the effect of high-pass filtering, e and f were generated using high-pass

filtered phase images.


142

Figure 6.4 Theoretically predicted CNRs of cylinders with different susceptibility values.

Figures in different rows were generated using different χ2 values, while figures in

different columns were generated using different χ1 values. a) χ1=0, χ2=1ppm; b) χ1=3σχ,

χ2=1ppm; c) χ1=0, χ2=0.45ppm; and d) χ1=3σχ, χ2=0.45ppm.

When a high-pass filtered phase was used, the optimal choice of n was not affected much

for =0 and = 0.45ppm (Fig. 6.3.e), but was slightly bigger for and =

0.45ppm for all the cylinders except for the smallest one (Fig. 6.3.f). For both = 0

and , the maximal CNR was reduced for bigger cylinders, when the high-pass

filter was used. This is also partly evident in Fig. 6.2. The behavior in the case when the

high pass filtered phase was used, agreed with the pattern observed in the theoretically

predicted CNRs for objects with low susceptibility in Fig. 6.4. Given the simulated results


143

shown in Fig. 6.3, we chose = 0.45ppm in the in vivo studies for a consistent choice of

n for the maximal CNR.

In vivo studies

The local CNRs of the two veins, 1) the right septal vein and 2) the left internal cerebral

vein, normalized by the SNRs, are plotted in Figs. 6.5 and 6.6 for all the in vivo datasets.

The normalized CNRs in the SWI images and in the original magnitude images, as well

as the SNRs in the original magnitude images (from the background tissues) are shown in

Table 6.2. Compared to the original magnitude images, SWI improves the local CNR of

the right septal vein in the anisotropic data, but not in the isotropic data. The CNR of the

left internal cerebral vein is not improved in SWI in either isotropic or anisotropic data,

due to the amplification of the noise in the background tissue region. Compared with the

CNRs in magnitude images, the local CNRs in tSWI were improved by roughly a factor

of 2 in both isotropic and anisotropic cases for = 0. Compared with conventional SWI,

the CNRs were improved by a factor of greater than three for datasets with isotropic

resolution and greater than 30% for datasets with anisotropic resolution in tSWI. The

local CNRs were further improved when . Considering all cases, when = 0

was used, n = 2 was a reasonable practical choice for both isotropic and anisotropic

datasets; when was used, n = 4 was a reasonable choice for isotropic datasets

and n = 8 for anisotropic datasets.


144


vein (b, d, and f) from different datasets with isotropic resolution. a, b, c and d were




Dataset 2 with isotropic resolution, when different data processing methods were used for

susceptibility mapping (see Fig. 6.7 for examples of the tSWI images).


145


vein (b, d, and f) from different datasets with anisotropic resolution. a, b, c and d were




Dataset 2 with anisotropic resolution, when different data processing methods were used

for susceptibility mapping (see Fig. 6.7 for examples of the tSWI images).


146

When SHARP along with an iterative algorithm (8) was used to generate susceptibility

maps, the CNRs of the two veins of interest in the corresponding tSWI image were

improved, as shown in Figs. 6.5.c, 6.5.d, 6.6.c and 6.6.d. For isotropic resolution, at n=2

with = 0, the relative improvements for the left internal cerebral vein and the right

septal vein were 23% and 14%, respectively. For anisotropic resolution and at n=2, the

relative improvements for the two veins were less than 5%. However, this improvement

in CNR was more significant for grey matter structures than for veins, as can be seen

from Fig. 6.7.d and 6.7.h.

In Fig. 6.7, we compare the tSWI and SWI minimum intensity projections for both the

isotropic and anisotropic cases. The tSWI appears to have higher CNR than the

conventional SWI in both isotropic and anisotropic data. For tSWI, isotropic data

provided a better delineation of the venous structures, compared to the anisotropic data.

This is consistent with the results shown in Figs. 6.5 and 6.6, in which the normalized

maximal CNRs are higher for the isotropic data than those for the anisotropic data. When

was used, the visibility of some tiny veins and the grey matter structures was

reduced compared to the case when = 0 was used.


147

Figure 6.7 Comparison between minimal intensity projections (mIP) of tSWI and SWI

data over 16mm for isotropic (top row) and anisotropic data (bottom row) for Dataset 2.

For b, c, f and g, susceptibility maps were generated using homodyne high-pass filtering

and thresholded k-space division; while for d and h, susceptibility maps were generated

using SHARP and geometry constrained iterative algorithm. a) isotropic SWI mIP; b)

isotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2); c) isotropic tSWI mIP (χ1=3σχ,

χ2=0.45ppm, n=4); d) isotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2); e) anisotropic SWI

mIP. f) anisotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2). g) anisotropic tSWI mIP (χ1=3σχ,

χ2=0.45ppm, n=8). h) anisotropic tSWI mIP (χ1=0, χ2=0.45ppm, n=2).


148

As an example of this process with = 0, = 0.45ppm and n = 2, Fig. 6.8 shows a

case that demonstrates the problems with the conventional SWI processing: one of the

veins has a trajectory roughly at the magic angle (54.7°) with respect to the direction of

the main magnetic field B0. The black arrow shows the vein which is clearly seen in the

tSWI (Fig. 6.8.f). In SWI (Fig. 6.8.e), the vein actually shows a dark structure which is in

fact more associated with its edges. This makes the veins appear much bigger in the SWI

than in the tSWI data, as can be seen from the minimum intensity projections (mIPs) in

Figs. 6.8.g and 6.8.h. This non-local phase information used in SWI can lead to an

inaccurate estimation of the geometry of microbleeds, as demonstrated in Fig. 6.9. In this

TBI case, tSWI has more faithfully represented the microbleeds.


149

Figure 6.8 A sagittal view showing a vein near the magic angle (54.7° relative to the

main magnetic field) as indicated by the black arrows. a) Phase image (from a left-handed

system) showing effectively zero phase inside the vein, with outer field dipole effects also

visible; b) susceptibility maps showing the vein as uniformly bright; c) susceptibility

weighting mask obtained from the phase image (n=4); d) susceptibility weighting mask

obtained from the susceptibility maps (χ1=0, χ2=0.45ppm, n=2); e) SWI showing

unsuppressed signal inside the vein; and f) tSWI showing a clear suppression of the vein

even at the magic angle. g) mIP of SWI in the sagittal direction. h) mIP of tSWI in the

sagittal direction. Note the vessels near the magic angle are now well delineated in the

tSWI data.


150

Figure 6.9 Sagittal views of SWI (a and c) and tSWI images (b and d) in a TBI case. The

microbleeds appear much bigger on the SWI images than on the tSWI images, as

indicated by the white arrows. This is due to the non-local phase information used in the

conventional SWI weighting mask. For better visualization, the images were interpolated

in through-plane direction from a resolution of 0.5mm×0.5mm×2mm to 0.5mm isotropic

resolution.

6.4 Discussion

Quantitative susceptibility mapping offers an additional means to recognize veins and

microbleeds and other tissues with high iron content as phase imaging does. However, the

phase images are dependent on each object’s shape and orientation while the

susceptibility values of the structures are not, at least in principle. Therefore, to produce


151

better susceptibility weighted images, we have investigated the use of susceptibility maps

for the masking process.

There are a number of key observations that can be made from the data presented herein.

First, the results presented in this paper demonstrate that the object shape and orientation

can be reasonably accounted for by using susceptibility maps and, hence, the inability of

SWI processing to enhance veins at different orientations can be overcome. Besides, the

blooming artefact due to the dipolar phase of microbleeds in conventional SWI was

avoided. This leads to potential applications of this technique, for example, the evaluation

of microbleeds in TBI studies. Nonetheless, the blooming artefact also enables the

visualization of small or even sub-voxel structures using conventional SWI (6). In this

case, due to the severe partial volume effects, the susceptibility values will not be

accurately estimated and it is expected that tSWI will lead to similar results as

conventional SWI. Second, tSWI can be used to process isotropic data, whereas SWI

processing has relied on anisotropic data for its best results due to the direct use of phase

information which is orientation dependent (6). In the past, SWI data have usually been

collected with anisotropic resolution with 2mm slice thickness(6). However, modern

segmented echo planar approaches are becoming viable and one expects to see more data

being collected with isotropic voxel sizes (19,20). The high isotropic resolution also helps

to reduce the error caused by the partial volume effects in susceptibility quantification,

and thus leads to improved quality for tSWI. Note that, high image resolution will also

lead to lower SNR within a given image time and thus lower CNR. Generally speaking,

tSWI is most advantageous with the isotropic datasets. Third, the use of a susceptibility


152

mask is not restricted to the paramagnetic venous blood, but it could also be designed to

study the diamagnetic materials (e.g., calcifications, which have negative susceptibility in

the susceptibility maps). Fourth, the effects of the upper and lower thresholds used in

creating the susceptibility masks have been studied for two reasonable values, and the

optimal number of multiplications, n has been determined. When the lower threshold was

set to zero, the fact that a continuous mask from zero to unity is generated makes it

possible to enhance contrast even in smaller veins, or larger veins that have had their

phase artificially suppressed by using the high-pass filter, or in structures that have lower

iron contents. The use of helps to avoid amplifying noise in regions of low

susceptibility and hence leads to a higher CNR. However, at the same time it can prevent

small veins or structures with very low susceptibility from being enhanced. In addition,

different datasets require different optimal n values. To avoid this problem, it may be

more practical to choose to be 0. We choose the upper threshold to be 0.45ppm, as

it corresponds to the theoretical susceptibility of venous blood when the oxygen

saturation is 70% and the hematocrit is 45%. Increasing this upper threshold may lead to a

slightly larger value for the optimal n when the susceptibility value of the vein is much

smaller than . In most cases, n = 2 gave optimal results, for = 0 and = 0.45ppm.

In order to capture smaller veins or structures with lower susceptibility values such as the

basal ganglia, a slightly larger n can be used for either isotropic or anisotropic datasets

(21). The lower susceptibility values are due to a combined effect of partial volume and

high-pass filtering. Fifth, the predicted CNRs slightly deviate from measured CNRs in

simulations, as the mean value of the susceptibility mask W in the background reference


153

region is slightly less than 1. As a result, this creates slight differences between the

prediction and measurements at large n values (n > 4). Lastly, conventional SWI uses

phase information which is dependent on echo time and usually a relatively long echo

time is used in SWI data collection. Although the phase mask could be redefined as a

function of echo time to accommodate the loss of phase information as echo times are

reduced, no such modification needs to take place for tSWI since the susceptibility map

does not change with echo time. However when echo times are reduced the phase SNR

used to generate the susceptibility map will decrease. On the other hand, if echo times

become too long, phase aliasing occurs and the apparent size of the vessel will increase.

Thus, tSWI makes the use of short echo times possible as long as the SNR is high enough

to create a reasonable estimate of the local susceptibilities. The selection of a shorter TE

has several major advantages, including reducing background field induced phase

artefacts, shorter scan time and better overall image quality.

There are several limitations to this method. First, we are using susceptibility maps

generated from a single orientation dataset to create the mask for tSWI. These

susceptibility maps can have streaking artefacts which are caused by the singularities in

the inverse kernel (7–15). The streaking artefacts could permeate the tSWI data causing

artefacts which did not exist before or decrease the CNR of grey matter structures. Some

newer techniques such as nonlinear regularization (10,13,14) and iterative algorithms (8)

will reduce the streaking artefacts and the latter is particularly time-efficient. Another

common problem of the single orientation QSM method is the systematic under-

estimation or bias of the susceptibility. However, this can be compensated by the


154

thresholds used to generate the tSWI weighting masks. Second, we used the traditional

homodyne high-pass filter to remove the background phase artefacts in the in vivo data.

Even though homodyne high-pass filtering could be applied without phase unwrapping, it

leads to an underestimation of the susceptibility, especially for large objects. This can be

improved by using newly developed background field removal methods (9,22). But for

relatively small structures such as veins, homodyne high-pass filtering already gives

satisfying results if it is performed in the appropriate plane (see Chapter 3). Given the fact

that homodyne high-pass filtering is still being widely used, the proposed algorithm can

be directly added to the current SWI data processing scheme.

In conclusion, we have proposed a data processing scheme which we refer to as true SWI

or tSWI to generate SWI like images using susceptibility maps. This helps to avoid the

orientation dependence related problem in SWI, especially in data with isotropic

resolution and, in the future, possibly to allow the use of short TE SWI data collection.

This tSWI data provide better and more consistent visualization of the venous system and

thus have potential clinical applications in the study of neurodegenerative diseases.


155

References



2. Mittal S, Wu Z, Neelavalli J, Haacke EM. Susceptibility-Weighted Imaging:

Technical Aspects and Clinical Applications, Part 2. Am. J. Neuroradiol.

2009;30:232–52.








6. Xu Y, Haacke EM. The role of voxel aspect ratio in determining apparent vascular

phase behavior in susceptibility weighted imaging. Magn. Reson. Imaging

2006;24:155–60.



2010;32:663–76.









NeuroImage 2012;62:2083–100.



Med. 2009;62:1510–22.



conditioning the inverse problem from measured magnetic field map to

susceptibility source image in MRI. Magn. Reson. Med. 2009;61:196–204.




2012;59:2560–8.

14. De Rochefort L, Liu T, Kressler B, Liu J, Spincemaille P, Lebon V, et al.

Quantitative susceptibility map reconstruction from MR phase data using bayesian

regularization: Validation and application to brain imaging. Magn. Reson. Med.

2010;63:194–206.


156

15. Wharton S, Bowtell R. Whole-brain susceptibility mapping at high field: A

comparison of multiple- and single-orientation methods. NeuroImage 2010;53:515–

25.

16. Jain V, Abdulmalik O, Propert KJ, Wehrli FW. Investigating the magnetic

susceptibility properties of fresh human blood for noninvasive oxygen saturation

quantification. Magn. Reson. Med. 2012;68:863–7.


55.



2007;46:6623–35.

19. Xu Y, Haacke EM. An iterative reconstruction technique for geometric distortion-

corrected segmented echo-planar imaging. Magn. Reson. Imaging 2008;26:1406–14.

20. Zwanenburg JJM, Versluis MJ, Luijten PR, Petridou N. Fast high resolution whole

brain T2* weighted imaging using echo planar imaging at 7T. NeuroImage

2011;56:1902–7.

21. Gho S-M, Liu C, Li W, Jang U, Kim EY, Hwang D, et al. Susceptibility map-

weighted imaging (SMWI) for neuroimaging. Magn. Reson. Med. 2013; doi:

10.1002/mrm.24920.



NMR Biomed. 2011;24:1129–36.


157

Chapter 7 Quantitative Susceptibility

Mapping of Small Objects using

Volume Constraints3

7.1 Introduction

The measurement of magnetic susceptibility offers an entirely new form of contrast in

magnetic resonance imaging (1-6). More specifically, susceptibility quantification has

already found applications in mapping out iron in the form of ferritin in brain tissues such

as the basal ganglia (1, 4, 6) and in the form of deoxyhemoglobin for measuring the

oxygen saturation in veins (1, 4). This new form of imaging may provide a means for

monitoring longitudinal changes in iron content in dementia, multiple sclerosis (MS),

traumatic brain injury (TBI) and Parkinson’s disease (PD). It may also be used to monitor

microbleeds which have been implicated in the progression of vascular dementia (7),

Alzheimer’s and other neurovascular disorders (8, 9).

3Most of the contents in this chapter are adapted from Liu S, Neelavalli J, Cheng Y-CN, Tang J, Haacke

EM. Quantitative susceptibility mapping of small objects using volume constraints. Magn. Reson. Med.

2013;69:716–23. Reprint under license.


158

One of the most recent susceptibility mapping methods is a Fourier based method (2, 3, 5,

10) which utilizes phase images. The accuracy of such a method depends on the volume

measurement of the object. For example, in order to quantify the susceptibility of a given

microbleed, usually the center and the radius of the microbleed have to be determined

(10-13). Alternate volume estimations of the microbleed from high resolution spin echo

images may overcome these limitations. With a gradient echo sequence, the apparent

volume of the object is increased due to what is commonly referred to as the “blooming”

effect, a signal loss around the object caused by T2* dephasing. This increased apparent

volume may be used to obtain an estimated susceptibility of the object while the product

of the apparent volume and the estimated susceptibility is much more robust and should

still provide a good estimate of the magnetic moment of the object.

The goal of this paper is to evaluate the quantitative accuracy of a Fourier based

susceptibility mapping method when it is applied to small structures, and to show that: 1)

an accurate estimate of the magnetic moment is possible using multi-echo gradient echo

imaging; and 2) the accuracy of the effective susceptibility can be improved using the

magnetic moment when an estimate of the true volume is available. For validation, we

used a gel phantom with air bubbles and glass beads to mimic the clinical situation of

microbleeds. The method illustrated here does not depend on the susceptibility value or

the size of the object.


159

7.2 Theory and Methods

Current susceptibility mapping methods are based on the relation between the

susceptibility distribution and the magnetic field variation in the Fourier domain (1-6; 10):

( ) [7.1],

where ( ) is the Fourier transform of the magnetic field variation , is the

Fourier transform of the susceptibility distribution , and is the Green’s

function

[7.2],

assuming that the main field direction is in the z direction. Susceptibility quantification is

an ill-posed inverse problem, due to zeros in the Green’s function along the magic

angle in the k-space domain. As a result, regularization is required. In this study, we

applied the regularization procedure described in a previous study (1) in which the

intensity of the inverse of is reasonably attenuated when the absolute value of

is below a threshold value, th. The selection of this threshold is a trade-off between the

susceptibility-to-noise ratio of the reconstructed susceptibility map and the accuracy in

susceptibility quantification (1, 6). The threshold value was chosen to be 0.1 in this study.

Although ideally is the sought after parameter, when reduced resolution or T2* effects

confound a clean measurement of the object’s volume, it is more appropriate to

investigate the associated magnetic moment (or, equivalently, the total or integrated


160

susceptibility weighted by the voxel volume (12)). To see why this is the case, consider a

sphere with a susceptibility difference , the induced magnetic field at point

outside the sphere is given by (14):

( )

[7.3]

where , is the susceptibility inside the object, is the

susceptibility outside the object, is the radius of the sphere, is the distance from the

point to the center of the sphere, and is the angle between the point P and the

main field direction. For simplicity, the meaning of susceptibility in this paper will be

taken to be rather than or . Eq. 7.3 also indicates that the product is

independent of TE, where is the true volume of the sphere. Since the

magnetic dipole moment of the spherical object is given as (14):

[7.4],

when . For simplicity we refer to the product as the magnetic moment in this

study. The phase value at a particular echo time (TE) is given in a right handed system by:

[7.5].

The susceptibility may be quantified using the phase information, if the true volume

(V) of the object is known. Otherwise, the magnetic moment ( ) may be found. Since

gradient echo images lead to a dephasing artefact and the object appears larger than its

actual size, we defined an apparent volume V′ and assuming that the susceptibility of this


161

larger object can be accurately quantified, the magnetic moment could still be accurately

calculated. Expressed symbolically, an estimated susceptibility value can be

calculated from the Fourier based method using Eqs. 7.1 to 7.5. The quantity

provides an estimate of the magnetic moment. Finally, the true susceptibility can be

calculated using the following equation:

[7.6].

In this study, we use three volume definitions. The first one is the true volume V. The

second one is the apparent volume V′, which is used in estimating the magnetic moment.

The apparent volume is related to the signal loss due to T2* dephasing and is determined

from gradient echo magnitude images, as described later. The last one is the spin echo

volume Vse, which is measured from the spin echo images. This volume is used as an MR

based estimate of the true volume of the air bubble or glass bead. We used simulations

and multi-echo gradient echo images of a gel phantom containing air-bubbles and glass

beads of varying sizes to test Eq. 7.6. While glass beads can be considered as almost

perfect spheres, air bubbles are closer to the clinical situation of variable shaped

microbleeds.

Simulations

To evaluate validity of Eq. 7.6 for susceptibility calculation of small objects, we

simulated magnitude and phase images of 4 spheres with different radii at 21 different

TEs (from 0 ms to 20 ms, with a step size of 1 ms). In each simulation, the sphere was

placed in the center of a 10243 matrix with complex elements. The radii of four spheres


162

tested, within this 10243 matrix, were 32, 48, 64 and 96 pixels, respectively. The

magnitude inside each sphere was set to 0 while the background magnitude was set to 300

to simulate intensities in the experimental data from the gel phantom. The phase images

of the spheres were generated according to Eqs. 7.3 and 7.5 with ppm. In order

to simulate Gibbs ringing as well as partial volume effects seen in actual MR data, a

process simulating the MR data sampling was used. Complex images generated in each

10243 matrix were Fourier transformed into k-space. The central 32

3 region was selected

from k-space and was inverse Fourier transformed back to the imaging domain generating

low resolution data containing both Gibbs ringing and partial volume effects. The radii of

the four spheres became 1, 1.5, 2 and 3 pixels respectively in this final 323 volume. White

Gaussian noise was then added to the real and imaginary channels of the complex data in

the image domain such that the SNR in resultant magnitude images was 10:1.

Susceptibility and the magnetic moment values were quantified for each of the spheres at

all echo times and errors associated with these measurements were evaluated.

Phantom experiments

A gel phantom, containing 14 small air-bubbles and 9 glass beads of varying sizes, was

imaged at 3T (Siemens VERIO, Erlangen, Germany) using a five-echo 3D gradient echo

sequence. The echo times (TEs) were 3.93ms, 9.60ms, 15.27ms, 20.94ms and 26.61ms.

Other imaging parameters for the gradient echo sequence were: repetition time (TR) 33ms,

flip angle (FA) 11º, read bandwidth (BW) 465 Hz/pixel, voxel size 0.5×0.5×0.5mm3, and

matrix size 512×304×176. A multi-slice 2D spin echo dataset was also collected with FA

= 90º, TR = 5000ms and TE = 15ms and with the same field of view (FOV), BW,


163

resolution and matrix size as in the gradient echo dataset. This is to maintain a one-to-one

correspondence of the spin echo with the gradient echo images of the phantom. To ensure

that the field perturbation measured in the phase images is the actual perturbation profile

from the gel phantom, we first performed shimming using a spherical phantom

immediately before performing the imaging experiment. Manual shimming was

performed on the spherical phantom, to a spectral full width at half maximum of 13 Hz

and the shim coefficients were noted. The same shim settings were used while imaging

the gel phantom to ensure that field perturbation profile due to the presence of the

phantom in the magnet is not influenced by any additional shimming.

For the construction of the phantom, an agarose gel solution was prepared with an 8%

concentration by weight and poured into a cylindrical container. In the lower portion of

the container, the gel was first filled to 1/3 the height of the cylinder and 9 glass beads of

various sizes were embedded in the gel. The true diameter of the glass beads was roughly

measured using calipers before the glass beads were put into the gel solution. Specifically,

4 glass beads were 2mm in diameter, 3 glass beads were 3mm in diameter, 1 glass bead

was 5mm in diameter, and the largest glass bead was 6mm in diameter. The phantom was

allowed to cool so that the gel solidified and properly engulfed the glass beads. Rest of

the prepared gel solution was then poured into the cylindrical container and variable sized

bubbles were injected by pumping various amount of air into the gel using an empty

syringe (two smallest air bubbles were excluded from this study, due to the limitation in

volume estimation of small objects. Details are provided in later sections). The theoretical

susceptibility difference between air and water is known to be 9.4ppm and will be used to


164

compare with the measurements from our method. For glass beads, the susceptibility

values were measured independently in a former study to be −1.8 +/- 0.3ppm (15).

First, in order to identify air bubbles and glass beads in the collected MR data, binary

masks from magnitude data were used. The intensity variation in the magnitude images

caused by the RF field inhomogeneity was first removed using a 2D quadratic fitting,

before the binary masks were created. A reasonably uniform magnitude intensity profile

across the phantom was obtained after this intensity correction. The binary masks were

created by local thresholding of the corrected magnitude images (11). First, a relatively

strict threshold is used to pick only the voxels where the signal is less than 50% of the

signal-to-noise ratio (SNR) in the gel away from the air bubbles or glass beads, since both

air bubbles and glass beads have much lower intensities than the intensity of the

surrounding gel. Next, the mean ( ) and standard deviation ( ) were

calculated for a cubic 21×21×21 VOI for each bubble or glass bead. A voxel roughly at

the center of the bubble or glass bead was first chosen to center this 213

voxel window.

The voxels picked up in the first step were excluded in the mean and standard deviation

calculation. If a neighboring voxel has intensity lower than , it was

regarded as a voxel belonging to air or glass bead. For the high SNR data used here, β

was empirically chosen to be 4 to separate air bubbles and glass beads from gel.


165

Susceptibility Quantification

In order to reduce the background field or phase variation, a forward modeling approach

was used to estimate air/gel-phantom interface effects (16). The phase processing steps

were as follows:

i. The original phase images were first unwrapped using the phase unwrapping tool,

PRELUDE, in FMRIB Software Library (FSL) (17). With the geometry of the gel

phantom extracted from the magnitude images at the shortest TE (3.93ms in this

study), the background field effects were reduced by fitting the predicted phase to

the unwrapped phase by a least squares method. An additional 2D quadratic fitting

was added in order to remove the induced phase due to eddy currents.

ii. The phase value inside a particular air bubble/glass bead (where the binary mask

is 1) was set to the mean phase (essentially zero) from the local 9260 voxels. This

is due to the fact that the phase inside a sphere is theoretically zero and the

nonzero phase is induced by the remnant background field variation as well as

Gibbs ringing. This step also determines the apparent volume (V′) from magnitude

images.

iii. At each echo time, a 160×160×87 voxel volume was cropped from the original

phase images. This volume was selected because it covers most of the air bubbles

and glass beads while voxels near the edge of the gel phantom were excluded. The

selected volume was then zero-filled to a 512×512×256 matrix.

iv. Susceptibility maps were generated using a threshold based approach described

previously in (1). The mean ( or ) and standard deviation (


166

or ) of the susceptibility values of air bubble (or glass bead) were

measured, taking into account the background susceptibility of the gel.

Measurements were obtained in the following manner: the background mean

( ) and standard deviation ( ) of the local gel susceptibility value around

each bubble or glass bead was first calculated from the 213 voxel region centered

at each of the bubble/bead. Within this 213 volume, the voxels belonging to the

bubble or glass bead, as determined by the binary mask, were excluded for this

background mean and standard deviation calculation. Once these measures were

obtained, for susceptibility of air bubbles, only voxels with susceptibility values

higher than were used for calculation purposes; while for glass

beads, only voxels with a susceptibility value lower than were

used. This process assumes that the noise in the susceptibility maps follows a

Gaussian distribution, and the susceptibility of a voxel consisting of air or glass is

statistically different from a voxel consisting of gel. The change in sign is due to

the fact that the air bubbles are paramagnetic relative to the gel while glass beads

are diamagnetic. To account for the baseline shift caused by remnant field

variation, the susceptibility of the air bubble (or glass bead) was taken as

(or ).

Comparison with recent data processing algorithms

In addition to the above described data processing steps, the phantom data were also

processed using newer data processing algorithms to further reduce the streaking artefact

in the susceptibility maps and to improve the accuracy of susceptibility quantification.


167

The background phase was removed using SHARP with radius 6 pixels, regularization

threshold 0.05 (18). Susceptibility maps were generated using the k-space/image domain

iterative algorithm (19). The geometries of the air bubbles and glass beads were extracted

from magnitude images at different TEs. In each iteration step, the regions outside the air

bubbles or glass beads were set to 0, while the regions inside the bubbles and beads were

smoothed using an edge-preserving averaging filter. Then the cone of singularities in the

original k-space was updated. The phase inside the air bubbles and glass beads was still

set to zero initially, and was updated using the predicted phase from forward calculation

using the susceptibility maps in each iteration step. The susceptibilities of the air bubbles

and glass beads were measured in the same way as described in the previous section.

Volume Measurement

The apparent volume of the air bubble or glass bead was determined from the binary

masks directly, i.e., by counting the number of voxels inside the air bubble or glass bead.

On the other hand, the spin echo volume is measured utilizing the "object strength" notion

proposed by Tofts et al (20), in which the total intensity is measured for a particular

volume of interest (VOI). For a volume composed of two types of tissues, a and b, the

total intensity can be expressed as

[7.7],

where is the total intensity, “ ” and “ ” are the intensities of the voxels containing

purely tissue “ ” or tissue “ ”, respectively. The total number of voxels in this volume of

interest is denoted by “ ”, and the number of voxels occupied by tissue “ ” is denoted by


168

“ ”. Consequently, the number of voxels occupied by tissue “ ” can be expressed as

.

By varying the size of the VOI, the total intensity is linearly dependent on the number of

voxels in the VOI. While " " can be determined as the slope in the fit to Eq. 7.7, can

be calculated from the intercept if “ ” is given ( may not be an integer as partial

volume is included). In this study, " " corresponds to the intensity of a voxel composed

purely of gel, while " " corresponds to the intensity of a voxel composed purely of air or

glass. For a relatively large air bubble or glass bead, " " is dominated by the thermal

noise, which can be approximated as , where is the measured

standard deviation of the gel region in the magnitude images (21). For an air bubble or

glass bead with a radius generally less than 3 pixels, " " is a combination of thermal

noise and Gibbs ringing. To best account for these fluctuations, " " is calculated as:

{

[7.8]

where and are the measured mean values inside the bubble and glass

bead, respectively; and are two weighting factors. Based on our simulations

(explained below), and were empirically determined from simulations to be 0.4

and 0.6 respectively, to minimize the error in estimation of the true volume.


169

Error in Volume Measurement

Although a regression method is used to measure the spin echo volume, it is still affected

by partial volume effects, Gibbs ringing as well as random noise. The simulated

magnitude images at TE=0 were used to mimic spin echo magnitude images and to study

the error in spin echo volume estimation. In addition, to examine the stability of this

method due to thermal noise, the volume measurement evaluation was performed 10

times for each simulated sphere, with independently generated random noise for each of

these simulations. The errors were determined by comparing the measured volume with

the true volume. Note that, this error estimation does not apply for the apparent volume

which is determined directly from the binary masks.

7.3 Results

Simulations

Magnetic moments for simulated spheres were calculated with the measured

susceptibilities and the apparent volume for each sphere at a given echo time. The results

across different TEs are shown in Fig. 7.1. The measured volumes at different TEs were

normalized to the volume at the longest TE, while the measured magnetic moments were

normalized to the true magnetic moment, which is the product of input volume (i.e., the

true volume) of the sphere and the input susceptibility (true susceptibility) 9.4 ppm. The

normalized magnetic moment is roughly a constant for all spheres. However, for the

sphere with a radius less than 2 pixels, the magnetic moments measured in the short TE

range have more fluctuations than those measured at longer TEs. In addition, the

magnetic moments are under-estimated for all spheres. The mean normalized magnetic


170

moments were measured as: 0.85±0.04 (radius=1 pixel), 0.82±0.05 (radius=1.5 pixels),

0.81±0.03 (radius=2 pixels) and 0.81± 0.02 (radius=3 pixels).

After the magnetic moments were obtained, the susceptibility values were corrected using

the actual known volume (i.e., true volume) using Eq. 7.6. Specifically, the corrected

susceptibilities are: 7.95±0.38ppm (radius=1pixel), 7.70±0.43ppm (radius=1.5pixels),

7.62±0.27 (radius=2pixels), and 7.63±0.21 (radius=3pixels). There is still a 15% to 19%

under-estimation in the averaged susceptibility after attempting to correct the volume of

the sphere.

To evaluate the stability of the volume measuring method, we carried out 10 simulations

for each sphere at TE=0. The means and standard deviations of the percentage errors

relative to true volume for each sphere are: 18.02±27.26% (radius=1pixel), 1.89±12.18%

(radius=1.5pixels), 3.67±8.91% (radius=2pixels) and 2.09±2.54% (radius=3pixels). The

algorithm failed to quantify, in two of the 10 simulations for sphere with radius of 1pixel.

Larger errors and more variations of the volume measurements were seen in spheres with

radii less than 2 pixels. For the sphere with a radius of 3 pixels, the error in the volume

estimation appears to be within 5% using the proposed method. As can be expected, when

the object radius is only 1 pixel, the volume measurement becomes unstable.


171

Figure 7.1 Apparent volume normalized to the volume at TE = 20 ms (first column),

measured susceptibility (second column), and normalized magnetic moments (third

column) measured at different TEs of four different spheres. The dashed lines in the

second column (b, e, h and k) indicate the true susceptibility 9.4 ppm. For each sphere,

the effective magnetic moments were normalized to the true effective magnetic moment.


172

Phantom experiments

A total of 14 air bubbles and 9 glass beads were examined in the phantom data. The

measured spin echo volumes of the glass beads and air bubbles are shown in Tables 7.1

and 7.2. In these two tables, the glass beads as well as air bubbles are sorted based on

their spin echo volumes, from small to large objects. The diameters of these glass beads

calculated from their spin echo volumes agree reasonably well with their physically

measured diameters, as shown in Table 7.1. Also note that, the error in volume

measurement is unreliable for spherical objects with radii less than 1.5 pixels (14.13

voxels for the volume). The error is generally larger than 20%, as shown in the

simulations. Thus, the first two smallest air bubbles were excluded from the analysis.

Fig. 7.2 shows three orthogonal views of the susceptibility map of the largest glass bead

for the shortest TE and the longest TE. Fig. 7.3 shows the susceptibility maps obtained

using the newer data processing algorithms. Compared with the former results, the

streaking artefacts are reduced and the objects are more homogeneous.


173

Table 7.1 Spin echo volume (in voxels) and the diameter (in mm) calculated from spin

echo volume for each glass bead.

Bead

Measure

1 2 3 4 5 6 7 8 9

Spin Echo Volume 35.4 37.5 38.2 39.1 96.5 103.8 113.6 516.6 912.1

Spin Echo Diameter 2.0 2.1 2.1 2.1 2.9 2.9 3.0 5.0 6.0

Actual Diameter 2.0 2.0 2.0 2.0 3.0 3.0 3.0 5.0 6.0

Table 7.2 Spin echo volume (in voxel) of the 14 air bubbles.

Bubble 1 2 3 4 5 6 7

Volume 3.3 15.1 28.7 42.2 43.9 82.8 87.7

Bubble 8 9 10 11 12 13 14

Volume 92.7 118.2 170.5 238.6 288.7 322.7 897.2


174


(a, b and c) and TE=26.61ms (d, e and f). The main field direction is in “y” direction.

Glass bead No. 9 in Table 7.1 is pointed by the white arrows. The air bubbles are pointed

by the white dashed arrows.


(a, b and c) and TE=26.61ms (d, e and f), obtained using newer data processing

algorithms.


175

Using Eq. 7.6, the measured susceptibilities can be corrected with the volume estimated

from the spin echo images. These results were shown in Table 7.3. The results obtained

using the newer algorithms were shown in Table 7.4. Using the original data processing

methods (i.e. geometry based artefact reduction together with 2D quadratic fitting for

background field removal, and truncated k-space division for susceptibility mapping), the

mean of the corrected susceptibility values of the glass beads averaged over all the TEs

was −1.82±0.17ppm, which is within the range of the measured values in the previous

study (15). The mean of the corrected susceptibility values of the air bubbles was

6.66±0.85ppm. This is to be compared to the actual susceptibility of 9.4ppm. Using the

new data processing algorithms (i.e. SHARP for background field removal, iterative

SWIM for susceptibility mapping), the corrected average susceptibility of the glass beads

was -2.12±0.15ppm, and the corrected average susceptibility of the air bubbles was

7.35±1.13ppm.

The susceptibility values estimated using the newer data processing algorithms are plotted

as a function of echo time in Fig. 7.4. As shown in Fig. 7.4.a, the absolute susceptibility

values of both the air bubbles and glass beads decreases as TE increases, due to the over-

estimated volume. And the corrected susceptibility values are almost constant over

different TEs, indicating that the product of susceptibility and the measured volume, or

the effective magnetic moment, is also constant. After correction, both the distributions of

susceptibility of air bubble and glass beads became narrower, indicating reduced

uncertainties in the susceptibility estimates, as shown in Fig. 7.4.c and 7.4.d.


176


air bubbles and different TEs.

Glass Bead Air bubble

TE/ms Measured Corrected Measured Corrected

3.93 -1.50±0.07 -1.79±0.13 3.15±1.16 6.13±0.77

9.60 -1.07±0.32 -1.76±0.13 1.70±0.61 6.44±0.90

15.27 -0.86±0.32 -1.82±0.19 1.30±0.45 6.64±0.76

20.94 -0.68±0.28 -1.82±0.18 1.07±0.36 6.88±0.77

26.61 -0.59±0.25 -1.88±0.21 0.93±0.34 7.22±0.74


air bubbles and different TEs. These results were obtained using the new data processing

algorithms.

Glass Bead Air bubble

TE/ms Measured Corrected Measured Corrected

3.93 -1.73±0.06 -2.06±0.15 3.58±1.41 7.24±1.01

9.60 -1.29±0.35 -2.13±0.15 1.87±0.68 7.40±1.16

15.27 -1.02±0.38 -2.16±0.16 1.36±0.48 7.39±1.20

20.94 -0.79±0.34 -2.10±0.14 1.08±0.37 7.35±1.21

26.61 -0.68±0.29 -2.17±0.18 0.89±0.31 7.38±1.23


177

Figure 7.4 a) Originally measured susceptibility values at different TEs for glass beads

and air bubbles. b) Corrected susceptibility values. c) Distribution of the originally

measured susceptibility values. d) Distribution of the corrected susceptibility values.

After correction using the spin echo volume, the glass beads can be clearly distinguished

from air bubbles. These results were obtained using the new data processing algorithms.

7.4 Discussion and Conclusions

The susceptibility mapping technique using the regularized Fourier based method has

certain advantages over other methods, especially in terms of time-efficiency and

simplicity. However, it suffers from problems caused by the intrinsic singularities in the

inverse of the Green's function, as well as partial volume effects which disrupt the true

0 10 20 30

-2

0

2

4

6

8

10

12

TE / ms

Su

sce

ptib

ility / p

pm

0 10 20 30

-2

0

2

4

6

8

10

12

TE / ms

Su

sce

ptib

ility / p

pm

-5 0 5 10 150

5

10

15

20

Susceptibility / ppm

Co

un

ts

Glass beads

Air bubbles

-5 0 5 10 150

10

20

30

40

50

Susceptibility / ppm

Co

un

ts

Glass beads

Air bubbles

0 5 10 15 20 25 30

-2

0

2

4

6

8

10

12

14

TE / ms

Susceptibili

ty /

ppm

Glass bead

Air bubble

a b

c d


178

phase behavior. For small objects, susceptibility quantification using the inverse method

(1) yields a significant under-estimation of the susceptibility. The increased apparent

volume at long TE can be utilized to create a larger virtual object, for which the true

volume can be more accurately measured and thus the Fourier based susceptibility

quantification gives a relatively smaller error for the magnetic moment. At this point, the

susceptibility close to the actual value can be extracted from the estimated magnetic

moment with an estimation of the true volume, either if it is known ahead of time, or it

can be estimated from a high resolution spin echo dataset.

Based on the discussions above, the error in the corrected susceptibility comes

from the estimated magnetic moment = and estimated volume (V). Through

error propagation, the error in the corrected susceptibility is given by:

√(

)

(

)

[7.9]

As can be seen from Eq. 7.9, the smaller the error in the estimated volume, the smaller the

error in the corrected susceptibility. This equation explains the error seen in the corrected

susceptibility of the air bubbles as well as glass beads.

In simulations, where the true volume is known, the remnant under-estimation in the

averaged corrected susceptibility ranges from 15% to 19%. Since there is no error in the

true volume, this error must be due to the error in the apparent volume measurement and

quantification due to the regularization process. The level of under-estimation is

related to the threshold value in the inverse of the Green’s function. A smaller threshold


179

leads to less under-estimation, but more streaking artefacts in the susceptibility maps. The

regularized Fourier based method, with threshold value of 0.1, can lead to an under-

estimation of around 13% for objects with radii larger than 3 pixels and even worse for

smaller objects (1, 6). This can be viewed as a systematic error, and can be reduced using

the k-space/image domain iterative algorithm, as shown in previous chapters.

In phantom studies, using high-pass filtered phase images and truncated k-space division

for generating susceptibility maps, the corrected susceptibility values of the air bubbles

have a maximum underestimation close to 44%, compared to the theoretical value 9.4ppm,

even after using the volume estimated from the spin echo data. This is essentially a

consequence of three factors: the error in the spin echo volume measurement, the signal

loss due to high-pass filtering, and the under-estimation of quantified using the

truncated k-space division method. To overcome the limitations related to volume

estimation, one has to go to high resolution images that can minimize volume

quantification error. However, the decreased SNR in high resolution spin echo images

may introduce additional variation/noise in the final volume results.

The accuracy in the susceptibility estimation can be further improved using more

sophisticated background field removal and susceptibility mapping methods. Using the

newer data processing algorithms, the background phase was better reduced. Particularly,

the streaking artefacts surrounding the air bubbles and glass beads were largely reduced

due to the use of geometry constrained iterative SWIM algorithm. Finally, the phase

values of the pixels inside the air bubbles and glass beads are not reliable, due to the low


180

signal inside these objects. This is solved by updating the phase values inside, using the

predicted phase, during the iterative SWIM algorithm. This helps to get a faster

convergence. All of these help to improve the accuracy in estimating the susceptibilities

of small objects with low signal inside.

There are a number of limitations to this study. Even though a forward calculation was

carried out to reduce the geometry induced field variation, remnant background field

variation still exists. To best account for it, the phase inside the spherical objects was set

to the local average phase. This also helps to reduce the large variation in susceptibility

estimate induced by Gibbs ringing and thermal noise. However, this phase correction

process is based on the assumption that the object of interest is a sphere. For non-

spherical objects, this phase correction process may lead to variations of magnetic

moment at different TEs. In addition, phase correction also creates a virtually larger

object. It is possible that the center of the created object deviates from the true center of

the original object of interest. This leads to additional errors even for spherical objects, as

seen from simulations. Thus, the phase inside the spherical object has significant effects

to this method. Theoretically, only when the center of a simulated large sphere coincides

with the original center of the sphere, and when the background phase value is 0, can we

obtain constant magnetic moment across different TEs. Hence, slight variation in object

definition from binary mask, which is used for phase substitution, can introduce

variations in magnetic moment values. This is the essential source of shape dependence of

the proposed method.


181

Although the estimated magnetic moments of the glass beads are almost a constant over

different TEs, as indicated by the corrected susceptibilities, the estimated magnetic

moments of the air bubbles are usually larger at a longer TE than at a shorter TE. This can

be understood by the fact that the air bubbles are not perfect spherical objects compared to

the glass beads. In fact, most of the air bubbles have ellipsoidal shapes, and any attempt of

phase correction inside the bubble based on the assumption of the spherical shape will

cause errors in the susceptibility measurement and thus lead to errors in the measurement

of the magnetic moments.

Generally speaking, for small objects which can be well approximated as spheres, the

theoretically expected errors in the estimated magnetic moment measurements are within

20% of the expected values and can be further reduced by adjusting the regularization

thresholds in the susceptibility mapping method. Practically, the errors might be larger

due to the limited knowledge of the true volume. While most small microbleeds can be

well approximated as spheres, the use of more accurate volume estimation methods has

the potential to reduce the error in susceptibility quantification of microbleeds.

In conclusion, we have shown that for very small structures, obtaining accurate magnetic

susceptibility values is limited by the errors in the volume estimations of these structures

and in the Fourier based method itself. Despite this inability to estimate the actual volume

of a small object accurately (whether it is an air bubble or microbleed), the estimated

magnetic moment is almost a constant over different TEs. This demonstrates that it is

possible to measure the magnetic moment at a longer TE when the apparent volume is


182

increased due to T2* dephasing. By measuring or knowing a priori the actual volume of

an object, it is possible to obtain a reasonable estimate of the susceptibility.

Acknowledgments This work is supported in part by National Institutes of Health via

grants NHLBI R01HL062983-A4 and NHLBI R21 HL 108230-A2. This work is also

supported in part by the Wayne State University Perinatal Research Initiative.


183

References


visualize veins and quantify oxygen saturation. Journal of Magnetic Resonance

Imaging 2010; 32(3):663-676.



conditioning the inverse problem from measured magnetic field map to susceptibility

source image in MRI. Magnetic Resonance in Medicine 2009; 61(1):196-204.

3. de Rochefort L, Brown R, Prince MR, Wang Y. Quantitative MR susceptibility

mapping using piece-wise constant regularized inversion of the magnetic field.

Magnetic Resonance in Medicine 2008; 60(4):1003-1009.

4. Schweser F, Deistung A, Lehr BW, Reichenbach JR. Quantitative imaging of intrinsic

magnetic tissue properties using MRI signal phase: An approach to in vivo brain iron

metabolism? NeuroImage 2011; 54(4):2789-2807.

5. Wharton S, Schäfer A, Bowtell R. Susceptibility mapping in the human brain using

threshold-based k-space division. Magnetic Resonance in Medicine 2010; 63(5):1292-

1304.


susceptibility mapping of brain tissue in vivo using MRI phase data. Magnetic

Resonance in Medicine 2009; 62(6):1510-1522.

7. Ayaz M, Boikov AS, Haacke EM, Kido DK, Kirsch WM. Imaging cerebral

microbleeds using susceptibility weighted imaging: One step toward detecting vascular

dementia. Journal of Magnetic Resonance Imaging 2010; 31(1):142-148.

8. Greenberg SM, Vernooij MW, Cordonnier C, Viswanathan A, Al-Shahi Salman R,

Warach S, Launer LJ, Van Buchem MA, Breteler MM. Cerebral microbleeds: a guide

to detection and interpretation. The Lancet Neurology 2009; 8(2):165-174.

9. Yates PA, Sirisriro R, Villemagne VL, Farquharson S, Masters CL, Rowe CC, For the

AIBL Research Group. Cerebral microhemorrhage and brain β-amyloid in aging and

Alzheimer disease. Neurology 2011; 77(1):48 -54.


and magnetic susceptibilities in MRI using a Fourier based method. Physics in

Medicine and Biology. 2009; 54(5):1169-1189.

11. McAuley G, Schrag M, Barnes S, Obenaus A, Dickson A, Holshouser B, Kirsch W.

Iron quantification of microbleeds in postmortem brain. Magnetic Resonance in

Medicine 2011; 65(6):1592-1601.


Microbleeds : Burden Assessment by Using Quantitative Susceptibility Mapping.

Radiology (2012); 262(1):269-78.

13. Cheng Y-C, Hsieh C-Y, Neelavalli J, Haacke EM. Quantifying effective magnetic

moments of narrow cylindrical objects in MRI. Physics in Medicine and Biology. 2009;

54(22):7025-7044.



http://www.ncbi.nlm.nih.gov/pubmed?term=%22Liu%20T%22%5BAuthor%5D

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Surapaneni%20K%22%5BAuthor%5D

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Lou%20M%22%5BAuthor%5D

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Cheng%20L%22%5BAuthor%5D

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Spincemaille%20P%22%5BAuthor%5D

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Wang%20Y%22%5BAuthor%5D


184

15. Hsieh C, Cheng Y, Tackett R, Kumar R, Lawes G, Haacke E. TH-D-304A-02:

Quantifying Magnetic Moments and Susceptibilities of Small Spherical Objects in

MRI. Medical Physics 2009; 36:2816.

16. Neelavalli J, Cheng YN, Jiang J, Haacke EM. Removing background phase variations

in susceptibility-weighted imaging using a fast, forward-field calculation. Journal of

Magnetic Resonance Imaging 2009; 29(4):937-948.

17. Jenkinson M. Fast, automated, N-dimensional phase-unwrapping algorithm. Magnetic


18. Schweser F, Deistung A, Lehr BW, Reichenbach JR. Quantitative imaging of intrinsic

magnetic tissue properties using MRI signal phase: An approach to in vivo brain iron

metabolism? NeuroImage. 2011 Feb 14;54(4):2789–807.



reconstruction approach. Magn Reson Med. 2013 May;69(5):1396–407.

20. Tofts PS, Silver NC, Barker GJ, Gass A. Object strength-an accurate measure for

small objects that is insensitive to partial volume effects. MAGMA 2005; 18(3):162-

169.

21. Gudbjartsson, H., and Patz, S. The Rician distribution of noisy MRI data. Magnetic



185

Chapter 8 Conclusions and Future

Directions

Quantitative susceptibility mapping (QSM) is a promising technique to study tissue

properties such as iron content and function such as oxygen extraction in vivo. In this

thesis, several technical advances in the QSM methodology have been made which will

accelerate the data processing and improve the accuracy of QSM.

First, the quality of QSM is largely dependent on background field removal. From the

results of simulations and in vivo data studies, the spherical mean value filtering (SMV)

based techniques are the most effective method, both in terms of the time-efficiency and

accuracy. However, the regions close to the edge of the brain are usually eroded in order

to avoid noisy pixels outside the brain. This is the main remnant problem of these SMV

based background field removal algorithms. For studies focused on the cerebral venous

oxygen saturation, the superior sagittal sinus (SSS) is of great interest. But the accuracy

of the extracted local phase information for the superior sagittal sinus is poor, since SSS

is sitting at the edge of the brain and may be partly removed by the brain extraction step.

Thus, it is critical to keep this structure using the binary brain mask and to keep the phase

at the edge of the SSS or even outside the SSS in the skull (1,2).


186

The Local Spherical Mean Value filtering (LSMV) algorithm proposed in this thesis

improves the robustness and efficiency for background field removal. It is particularly

time-efficient for processing data collected with double- or multi- echo. The main

advantage of this algorithm is avoiding global phase unwrapping and the reduction of

phase signal loss near the edges of the veins due to T2* signal decay. It can also be used

for processing of data collected with a single short TE. The local background field

removal also helps to reduce the effects of cusp artefacts/phase singularities, which are

typically caused by improper combination of multi-channel data.

Proper multi-channel phase data combination is critical for data collected using phased

array coils and parallel imaging techniques such as GRAPPA (3). Any improper

combination of the phase will lead to signal cancellation and cusp artefacts (4,5). For

regions inside the brain, the cusp artefacts can be avoided by correcting the linear

gradients and baseline differences of different channels. But for regions outside the brain,

such a simple correction may still lead to cusp artefacts (6). This is again due to the

difference in the phase references of different channels, , which contains both the coil-

sensitivity induced phase components and the conductivity induced phase components.

Thus, the ideal phase combination should have the removed from each channel before

combining all the channels’ data. This can be done by using either a reference scan or

reference echo (6). In the former method, usually the reference scan is performed using

coils with relatively uniform sensitivity profile. In the latter method, the TE of the

reference echo could be chosen such that it equals half of the TE of the main scan. Then

can be calculated using these two echoes using complex division (without phase


187

unwrapping). Removing these spatially varying effects will then lead to an improved

QSM reconstruction.

Second, QSM is essentially an ill-posed inverse problem. For solving the inverse problem,

the main difference in currently available algorithms is related to the handling of the

gradients/edges in the susceptibility maps. In order to suppress false gradients caused by

the streaking artefacts, the gradients estimated from magnitude images and phase images

or a combination of both are usually used as a priori information (7,8). Care should be

taken when imposing these a priori constraints. While a false edge detected from

magnitude or phase images will reduce the effectiveness in removing the streaking

artefacts in the susceptibility maps, a missed edge will lead to over-smoothing of the

susceptibility maps. For clinical studies, the processing time for these methods is also of

concern, especially when large numbers of datasets with large matrix size are involved.

The k-space/image domain iterative algorithm (iterative SWIM) is a promising method,

precisely because of its proper preserving of the edge information in the susceptibility

maps and its time-efficiency. As validated using simulated data and in vivo data, this

iterative SWIM algorithm is particularly effective for studies focused on measuring

venous oxygen saturation.

An improved version of the iterative SWIM algorithm, which utilizes the geometries of

the structures found in the susceptibility maps for solving the inverse problem, is

demonstrated in this thesis. Instead of using the geometries of the structures with high

susceptibilities, e.g., veins, as in the original iterative SWIM algorithm (9), in the


188

improved version proposed in this thesis, both the geometries of the veins and grey matter

structures (with relatively high susceptibilities) were extracted and utilized in solving the

inverse problem. This improves the accuracy of susceptibility estimation of both the veins

and the grey matter structures.

The accuracy of QSM is also dependent on the estimation of the volume, especially for

small objects (10). Longer TE gives higher phase values and potentially higher phase

SNR, but it also leads to more T2* signal decay and larger apparent volume of the object.

Any error in estimating the volume of the object will lead to error in estimating the

susceptibility of that object (10). On the other hand, the effective magnetic moment can

be used as surrogate to reflect the changes of the object of interest (10-12).

Furthermore, the susceptibility maps can also be combined with the magnitude images to

produce tSWI images (13,14). This helps to remove the orientation dependence of

conventional SWI data. These tSWI images provide improved delineation of the

geometries of veins and microbleeds and tSWI data processing can be easily incorporated

with the current SWI data processing scheme. Thus tSWI has great potential in studies of

traumatic brain injury (TBI) (13).

There are also a few limitations in the techniques presented in this thesis. First, all of the

techniques are for the data collected with single orientation. Orientation dependence of

the susceptibility values of the white matter structures has been reported in former studies,

and has been attributed to the myelin sheath surrounding the axons in the white matter

(15–18). This requires more sophisticated models to describe the relation between field


189

variation and susceptibility distribution (19–21). In addition to the susceptibility

anisotropy, nonlinear phase evolution was also observed and can be modeled using multi

water compartments (17). Both the orientation dependence and nonlinear phase evolution

are neglected in this thesis, since this thesis is focused on the deep grey matter structures

and veins (18). Second, the method of improving the accuracy of susceptibility

quantification using volume constraint was only demonstrated with a phantom, but

without any in vivo data. The main application of this method is quantification of cerebral

microbleeds or calcifications. For in vivo data, an estimate of the true volume of the

microbleeds from the spin echo data may not be available. If multi-echo data were

collected, it is possible to get an estimate of the true volume at TE=0 through

extrapolation using the measured apparent volumes at different TEs. Otherwise, the

effective magnetic moment could be used to study the longitudinal changes of the

microbleeds or calcifications (12).

Future directions include the following aspects. First, the performance of the iterative

SWIM algorithm for solving the ill-posed inverse problem in susceptibility mapping

could be improved. The effectiveness of the iterative SWIM algorithm relies on proper

extraction of the geometries of different structures of interest. Currently, the geometries

are extracted purely from susceptibility maps. The accuracy of the extracted geometries

could be potentially improved by incorporating the geometries found in the magnitude

and phase images (8). Second, the lost phase information near the edge of the brain, due

to erosion of the background field removal algorithm, should be recovered as much as

possible. This can be achieved by preserving the signal outside the brain. Previous studies


190

have already shown that it is possible to keep the phase in the skull region (22). This

provides more accurate boundary conditions for background field removal and thus

reduces the error of background field removal near the edges of the brain. Third, the

accuracy of susceptibility quantification could be further improved. Particularly, the

quantification of the susceptibility of veins is usually affected by partial volume effects,

especially for small veins. Future studies should aim to model and compensate for the

partial volume effects. Other future directions include: compensation for the signal loss

caused by homodyne high-pass filtering, in a way similar to the deconvolution step in

SHARP; incorporation of partial volume effects in the magnitude and phase images of the

3D brain model.

In conclusion, quantitative susceptibility mapping is a promising technique which has

many potential applications. It can be applied to most of the applications found by

susceptibility weighted imaging (5,23). It is also finding newer applications, given its

ability to quantify the source of the phase information and its independence of echo time

or orientation, such as tracking and quantifying iron labeled stem cells (24). The accuracy

of QSM is dependent on both the imaging acquisition and image post-processing. This

thesis has focused on two core steps of QSM post-processing: background field removal

and solving the inverse problem. While a new algorithm has been proposed for

background field removal, an improved method for solving the inverse problem has also

been demonstrated. Both of them contribute to a more sophisticated data processing

scheme for quantitative susceptibility mapping with improved effectiveness and accuracy.


191

References

1. Li L, Leigh JS. High-precision mapping of the magnetic field utilizing the harmonic

function mean value property. J. Magn. Reson. 1997 2001;148:442–8.

2. Zhou D, Liu T, Spincemaille P, Wang Y. Background field removal by solving the

Laplacian boundary value problem. NMR Biomed. 2014;

3. Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, et al.

Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn. Reson.

Med. 2002;47:1202–10.




diseases. NeuroImage 2008;39:1682–92.





Magn. Reson. Med. 2011;65:1638–48.




2012;59:2560–8.



NeuroImage 2012;62:2083–100.






Med. 2013;69:716–23.





192



Radiology 2012;262:269–78.

13. Liu S, Mok K, Neelavalli J, Cheng YCN, Tang J, Ye Y, et al. Improved MR

Venography Using Quantitative Susceptibility-Weighted Imaging. J. Magn. Reson.

Imaging 2013;

14. Gho S-M, Liu C, Li W, Jang U, Kim EY, Hwang D, et al. Susceptibility map-

weighted imaging (SMWI) for neuroimaging. Magn. Reson. Med. 2013;

15. Liu C, Li W, Johnson GA, Wu B. High-field (9.4 T) MRI of brain dysmyelination

by quantitative mapping of magnetic susceptibility. NeuroImage 2011;56:930–8.

16. Li X, Vikram DS, Lim IAL, Jones CK, Farrell JAD, van Zijl PCM. Mapping

magnetic susceptibility anisotropies of white matter in vivo in the human brain at 7

T. NeuroImage 2012;62:314–30.

17. Wharton S, Bowtell R. Fiber orientation-dependent white matter contrast in gradient

echo MRI. Proc. Natl. Acad. Sci. U. S. A. 2012;109:18559–64.

18. Rudko DA, Klassen LM, de Chickera SN, Gati JS, Dekaban GA, Menon RS.

Origins of orientation dependence in gray and white matter. Proc. Natl. Acad. Sci. U.

S. A. 2014;111:E159–167.

19. He X, Yablonskiy DA. Biophysical mechanisms of phase contrast in gradient echo

MRI. Proc. Natl. Acad. Sci. U. S. A. 2009;106:13558–63.

20. Liu C. Susceptibility tensor imaging. Magn. Reson. Med. 2010;63:1471–7.

21. Luo J, He X, Yablonskiy DA. Magnetic susceptibility induced white matter MR

signal frequency shifts-experimental comparison between Lorentzian sphere and

generalized Lorentzian approaches. Magn. Reson. Med. 2013;

22. Buch S, Liu S, Cheng Y, Haacke E. Susceptibility Mapping of the Sinuses in the

Brain by Preserving Phase Information in the Skull Using Short Echo Times. Proc.

21st Annu. Meet. ISMRM Salt Lake City Utah USA 2013;:2483.

23. Mittal S, Wu Z, Neelavalli J, Haacke EM. Susceptibility-Weighted Imaging:

Technical Aspects and Clinical Applications, Part 2. Am. J. Neuroradiol.

2009;30:232–52.

24. Ruggiero A, Guenoun J, Smit H, Doeswijk GN, Klein S, Krestin GP, et al. In vivo

MRI mapping of iron oxide-labeled stem cells transplanted in the heart. Contrast

Media Mol. Imaging 2013;8:487–94.

Technical Improvements in Quantitative Susceptibility Mapping · PDF fileTECHNICAL...

Documents

Transcript of Technical Improvements in Quantitative Susceptibility Mapping · PDF fileTECHNICAL...